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Complex Systems - Jean-Philippe Bouchaud - Lecture 7: Giant Components. Interactions, Choice Theory — Transcript

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0:00this conference this conference will now
0:03be recorded
0:04okay
0:06just started now
0:08if it's not too bad
0:11if not one minute is cut off
0:13Okay so
0:15you remember I was considering
0:18pre-like Networks so let me draw
0:21a tree like Network which is not
0:24necessarily regular uh okay though
0:27things like this and maybe there are
0:30occasional Loops but we'll neglect them
0:34and then I told you that there's a
0:37Criterion to know whether
0:39this network has a giant components or
0:42not
0:44in the language of physics this is
0:46called a percolation cluster
0:49and the idea was to map this problem
0:52onto a problem of population growth onto
0:57the the Carlton Watson model
1:00and so I tried to convince you that in
1:03this model in this in this framework you
1:06need to know
1:07the probability of that the node has K
1:11links are going from that node so this
1:15is called the degree distribution and
1:17then I told you that actually what is
1:19important for these type of questions is
1:22not pfk but qfk
1:27which
1:28I showed was given by kpfk divided by
1:37expectation of K according to p
1:40and the idea of qfk was not what is the
1:45probability that the randomly chosen
1:46node has K neighbors but what is the
1:49probability that knowing that I have
1:51chosen that node
1:52uh following a link from another node
1:55what is the probability that that chosen
1:58node has a neighbors and I try to
2:01convince you and of course this can be
2:03made rigorous through Bayes argument
2:07that you increase the probability for
2:10large or for nodes with a large degree
2:13simply because these nodes have a large
2:16number of friends so you're more likely
2:18to choose them anyway I'm not going back
2:20to this but in the end of this
2:23discussion I told you that if you think
2:26of
2:27propagation along this network as a kind
2:30of population growth uh problem then
2:33there's a an r0 a rate A reproduction
2:37rate of uh of the process that's that's
2:40growing I mean by analogy
2:43which tells you whether or not you will
2:46have a giant component and this r0 we we
2:51found that it was given by the average
2:53value of K squared
2:55so maybe I should stick to my
2:57notation
3:00from the past lecture and I know I will
3:03have to move my camera in a second
3:05so this was given by
3:07um
3:08and I'm leaving a space for a reason the
3:11expectation according to P of K squared
3:15divided by the expectation according to
3:18pfk
3:19minus one
3:22so then you move my camera
3:32okay
3:35and of course
3:37if r0 is less than one then we know that
3:41the population stops after a while and
3:44this means that you're actually on a
3:46finite cluster in the network
3:48whereas if r0 is greater than one then
3:51the process goes on forever and it means
3:54that you you are actually you have
3:56started on a giant a giant component or
3:59on the calculation cluster and so the
4:01Criterion was that this should be
4:04compared to one
4:05so if there is if this inequality is
4:09obeyed then
4:11there exists a giant cluster
4:13okay
4:16well now I want to know whether this
4:19giant cluster is robust is resilient or
4:22if it's going to crumble to Pieces as
4:25soon as I'm attacking it
4:28so you know you can think of this
4:31problem as the giant cluster needs to be
4:35destroyed because if you're in a in a
4:39model where you're actually modeling
4:41contagion of an of a disease along a
4:46social network then you don't want a
4:49giant component to exist because if the
4:50giant component exists it means that a
4:53finite fraction of the population will
4:56eventually get infected
4:58but if you think of this uh Network as
5:02for example
5:04um an electric grid or the internet I
5:08mean the physical Network then if you
5:11destroy the giant component this can
5:13have a dire consequences or the
5:16functioning of the electric Network for
5:19example and so you want this network to
5:21be robust against the failure of a few
5:25uh of its nodes okay so depending on the
5:29point of view you might be interested
5:30either in the resilience of the of the
5:33client component or on the fragility of
5:35the trying component but of course these
5:37are dual
5:39uh problems okay so now what I'm
5:42interested in is Imagine That I uh
5:46through an attack of this network which
5:49can be you know a physical attack trying
5:51to disturb uh the internet Network or
5:54the electric grid of a country or
5:57through a vaccination campaign I take
6:01down a few nodes okay so for example
6:04this node is deactivated this one is
6:07deactivated this one is deactivated
6:10and I call High
6:14the fraction of nodes that are down
6:18I is the fraction
6:22of
6:24disactivated nodes
6:36so again you know again really this is
6:39obviously a very topical uh subject you
6:41can just activate these nodes through
6:44vaccination if you think of a disease
6:46because you can think that this guy if
6:49he's vaccinated he's not going to be
6:51able to further uh transfer the illness
6:55from one neighbor to his other crowd of
6:59Neighbors
7:01okay so what is the r0
7:04in the presence of PSI
7:08so if you want I'm going to add a
7:12possible dependence on Phi here
7:17and if you think about what it means to
7:21destroy or to deactivate some nodes it
7:25means that from a given node you're
7:28actually reducing the number of children
7:30you remember that I I worked with
7:34analogy with the Galton Watson process
7:36and so you know you're coming from this
7:38node so this is the parent and this guy
7:42has a number of children and the
7:44question was how many children on
7:46average this node has and this was the
7:49reason for the minus one here because
7:51the average number of connection is this
7:54ratio and then you have to remove one
7:56because one of the of the link is
8:00actually uh
8:02taken already by
8:04um by the father son relationship okay
8:07this was last weeks or two weeks ago's
8:10argument so now I have to count the
8:12average number of children knowing that
8:15a fraction Phi of the nodes has been
8:17deactivated and so the argument is
8:20simply to multiply this quantity here by
8:241 minus five
8:26okay
8:28this is the reduction of the average
8:30number of children per node
8:33and so the generalized Criterion to know
8:36whether you have a giant component or
8:40not in the presence of this attack is
8:43very simple it's just this uh modified
8:46model read Criterion
8:49so modified
8:52Malloy read
8:58which again I've argued using a little
9:02bit of Van waving argument and this
9:04analogy with uh Kelson Watson but you
9:08know believe me you can actually do much
9:10better than that and through the same
9:13kind of calculation that would be needed
9:15to establish the Mallory Criterion to
9:19start with is is very simple to get this
9:23Factor one minus Phi on top of it
9:26okay
9:28so that's the result
9:30and in order to make it a little more uh
9:33visual
9:34I'm going to
9:36take an example and my example is going
9:39to be
9:40that I have a network
9:44which has
9:46um a degree distribution which is an
9:49exact parallel
9:51so I'm assuming that I'm constructing a
9:54network
9:55such that
9:57pfk is equal to a divided by K to the
10:01one plus u not only for large K but
10:04actually for all k
10:12um integer okay so this is my model I
10:15have a psk which is given by an exact
10:17parallel
10:18and a here
10:20is just a normalization so this is a one
10:23parameter model
10:30the only parameter is Mu so I have a
10:33certain a of mu here which is such that
10:36the sum over K of P of K is equal to one
10:39okay
10:41so from this TFK I can compute as a
10:45function of mu uh r0
10:48and therefore I can draw a phase diagram
10:52as a function of mu for what's going on
10:55in these types of models
10:58so what I'm going to draw
11:01is as a function of this time it's a mu
11:04and so I was going to draw
11:07P infinity and P Infinity if you
11:09remember is the probability that the
11:12randomly chosen node belongs to the
11:15giant component
11:17from
11:18that
11:20a node
11:23belongs to the giant component
11:27so
11:30you know you imagine that as Mu
11:33increases this distribution goes down uh
11:36very quickly and therefore it's going to
11:39be more and more concentrated for small
11:41values of K for large mu
11:44and therefore if it's weakly connected
11:47for large mirror you expect that there
11:50will not be any giant component in that
11:52case as you and as you decrease mu you
11:56have more and more what I call Hubs last
11:59time so nodes that are connected to many
12:01many other nodes and therefore you
12:03expect in this case that a joint
12:06component
12:07appears so as a function of mu
12:12let me
12:14indicate to special values
12:18then there's a transition as a function
12:21of U
12:22which looks like this
12:25um
12:26let me change color or maybe this is
12:28okay
12:29so
12:31above a value of mu which is roughly
12:34equal to 2.5
12:36then P Infinity is zero
12:40okay
12:43and then between this critical value of
12:45mu and mu equal one
12:49there's a curve that does like this
12:52and then below mule one p Infinity P
12:55Infinity is equal to one
12:58okay
13:01so what you see is that there are three
13:04phases in this model
13:06um in the summary of networks one when
13:09there is no giant component one when
13:11there is a giant component but it only
13:14occupies a a
13:16fraction below one of uh the nodes so
13:20not all nodes belong to in the giant
13:23component and then finally for Mu less
13:25than one then the distribution is so
13:28broad that everybody has to belong to
13:31the client component except the finite
13:34number of nodes in the large n limit
13:36okay now on the same graph
13:39I want to show what happens
13:42um
13:43if you start attacking this network with
13:47some fraction Phi of this activated node
13:50so what I'm going to draw is
13:535c what I'm going to call 5B which is
13:57the fraction of node Beyond which uh
14:00there is no
14:03um
14:04time component anymore
14:06so this R of Phi is given by this
14:08formula and what I'm going to draw is 5C
14:11which is defined by r0 of 5c
14:14equal one
14:17so as you increase by c as you increase
14:205 from this formula you see that you
14:23reduce r0 so if you start from a phase
14:26when there is some joint component
14:30then after a sun value Phi which I call
14:33Phi C the drawing component will be
14:36destroyed so that's the meaning of 5C
14:38and if you compute Phi C from again the
14:41same formula what you get is
14:47the curve that does like this the 5c is
14:50in blue
14:56so what it means is that of course if
14:58there's no giant component to start with
15:00then Phi C is zero you don't need to
15:03destroy anything the giant component is
15:05already uh not there
15:08then in this little region of the phase
15:12diagram between 2 and 2.5 then you you
15:15need to apply some fraction 5c to
15:19destroy the joint component but this 5c
15:21is less than one until you reach Miracle
15:252 and for Mu equal to actually if you
15:28want to get rid of the giant component
15:29you need to remove a fraction of side a
15:34fraction of node which is equal to one
15:36so this means that in this whole region
15:39the giant component is resilient
15:46very resilient because you really need
15:48to vaccinate nearly 100 of the
15:51population to get rid of the of of the
15:55transmission rate so this is due to
15:57Super spreaders and of course as again
15:59it's a very topical subject you know if
16:02uh the distribution of
16:05if the social networks are such that
16:07distribution of the degree is
16:10um narrow enough
16:11then ideas like Collective immunities
16:15and things like that that are simply
16:17related to r0 are valid and so as you
16:21know if in the in the naive model
16:24um the collective immunity which is 5c
16:28essentially is given by 1 minus one over
16:31r0 so that's in a naive model and uh so
16:35if r0 is three which is we believe the
16:39natural reproduction rate of the covid
16:41then one minus one over three is
16:44two-thirds and it means that two-thirds
16:47of the population need to be immune for
16:49the epidemic to stop so that's the
16:52number that we heard in the last year
16:56it's pretty high you know two-third of
16:59the population need to uh have been have
17:02had coveted for a natural immunity to
17:06um take place but you see that if the
17:09social network is contains super
17:11spreaders then it's much worse than that
17:13then you need to uh go beyond this uh
17:17naive elimination rate immunization rate
17:20to uh
17:22to get immunity
17:24Okay so
17:27this looks depressing however
17:31if you know that the network is the
17:34of that form
17:37then there's a trick to increase the
17:40vaccination efficacy and the trick is
17:43related to the argument that we use to
17:46get this result
17:48okay
17:50so you know if of course you don't know
17:53the degree of people you don't you
17:55cannot know how many friends a given
17:58person has
17:59so what you can do in order to improve a
18:02vaccination campaign is You Know It uh
18:06so the first idea would be to vaccinate
18:09people randomly but you can do a little
18:11better than that and in some cases much
18:13better than that by
18:16you know connecting some to someone
18:17contacting someone at random and then
18:20asking this person to name a friend who
18:23should be vaccinated
18:25and if you do that then you're exactly
18:28in the context of this selection uh
18:31process that I've described that
18:33transforms pfk into qfk
18:37and by this simple trick actually you
18:40can
18:41enhance tremendously a vaccination
18:44campaign and lower the density of buy
18:48that you need to
18:50uh to uh to reach to get rid of the
18:54client component and in a sense it's
18:56intuitive it's related to what I told
18:58you here I told you that the P Infinity
19:01is equal to 1 in this region because of
19:04the Hub so if you target The Hub
19:07then you're going to remove the giant
19:09component much more efficiently and the
19:11idea of targeting the hubs is a little
19:14bit what's contained in here if you
19:16don't know of course in electrical grid
19:18you can count the I mean it's it's a
19:22sometimes public information the the
19:24topology of the network so if you're a
19:27terrorist and want to destroy an
19:29electric grid you know obviously that
19:31you should Target
19:33um heavily connected nodes but this is
19:36easy to find in a population in order to
19:39detect who is connected to many people
19:41is much harder but this tricks this
19:45conditioning trick
19:46makes part of the of the way
19:50so I think it's an interesting
19:52you know it's an extreme extremely
19:54interesting example of this little
19:57Paradox that your friends on average
19:59have more friends than you have that you
20:01can turn to its head and use as a tool
20:05to improve
20:06a vaccination campaign
20:10okay so this is the end of my chapter
20:12three
20:13and so now I can move to uh chapter four
20:21interactions and Collective effects
20:25foreign
20:42so
20:44what I'm going to talk about is what I
20:47actually started my lectures with in the
20:50very first session I told you about
20:53crisis discontinuities and all these
20:56effects
20:57and so what I want to show you is a
21:01simple model where you can understand
21:03that depending on some parameters you
21:06can either have a slow Evolution smooth
21:09Evolution for example you know the state
21:12of optimism of a population or
21:17um you know the
21:18bullishness of people to buy the stock
21:21market or the trust in the banking
21:24system many examples like this so in
21:27some cases you can find that this this
21:31optimism say like this is evolving in a
21:35gradual smooth way but in other cases as
21:38we know there seems there seem to be
21:40tipping points Beyond which
21:43um there's a sudden breakdown of trust
21:46or of optimism there are crashes in
21:49financial markets and things like that
21:51and so we want to understand why in some
21:54cases things seem smooth and in other
21:56cases they seem abrupt
22:01um
22:02yeah I think it's very striking to see
22:04that in many social contexts
22:07you can have very sudden shifts in the
22:10in the way people perceive a problem
22:13it's very I guess that you know the last
22:1620 years in in terms of child abuse in
22:21France in particular it's very striking
22:23to see how you know child abuse was
22:26tolerated in the in even in the 90s and
22:29now it's completely unacceptable and so
22:32these these very these kind of pretty
22:34rapid change of uh of cultural norms in
22:38a in a society I think are extremely
22:40interesting to understand from a
22:43theoretical point of view
22:45Okay so
22:48what I'm going to present to you is
22:50what's called a random clizing model
22:54and it's of course a model that's been
22:56introduced in uh in a physical context
22:59so this is the physics
23:03of random magnets
23:15and the model in terms of describing the
23:18physics of random magnets has had a lot
23:19of successes in particular in describing
23:22what's called hysteresis loops and I'll
23:25go back to that in a second
23:27and so what I'm uh proposing here is to
23:30transpose the random female I think
23:32model to cases where people have to make
23:35binary decisions
23:37so I'm going to imagine that agent a
23:41agent I sorry
23:44has to make
23:48a binary decision
23:54which I'm going to call is called the
23:55spin
23:57so s i
23:58at time T is equal to plus 1 or -1
24:05and this binary decision as I said can
24:08be many things it can be
24:11buy or sell the stock market
24:19um
24:21it can be votes right or left
24:30it can be buying a cell phone I'm
24:33speaking about this because the
24:35there is data on how cell phones in the
24:3990s invaded the market so of course
24:43you're much too young to imagine the
24:44world without cell phones but in the 90s
24:48um at first it was not obvious at all
24:50that one should get the cell phone so
24:53you know imagine
24:55my cell phone because as I said
24:58there are there is data that I have
25:01analyzed myself on this problem but you
25:04can imagine more generally buy a new
25:06technology
25:18um
25:20tax evasion you can also imagine that
25:23whether you should Dodge taxes or pay
25:27your taxes
25:29um you know in some countries
25:31it's uh it's been a sport for a long
25:33time to dodge taxes but we can think of
25:37that as being part of the problem
25:40um
25:41in the US there's the gun problem you
25:44know should you buy a gun if you buy if
25:46you um if you live in the U.S or should
25:50you be for a gun control all these
25:53binary examples
25:57you can have in mind and you'll see why
26:00I'm putting all this in the same basket
26:03so the rule of the of the random field
26:05icing model that I want to transpose
26:07here
26:08let me write it down and then I'm going
26:11to comment what I'm trying to
26:13uh
26:15model so SI at time t plus 1 is the sine
26:21of the sum of three terms
26:24so it's a sign because this can only be
26:27plus or minus one so there's a term that
26:30I'm going to call
26:31Capital H of t
26:34which is common to everybody you see
26:37that here there's no I
26:40so that's what I'm going to call common
26:42information
26:44or public information
26:57then there's a term that's specific to
27:00each agent which I'm going to call H
27:02little I
27:04so this term here is I dependent but
27:08it's t independent it's static in time
27:12so this is the random field in the in
27:15the icing model so this is an
27:17idiosyncratic preferences
27:27okay so it means that you know depending
27:31on who you are depending on where you're
27:34born or what's your uh
27:37background and so on you may be more
27:40inclined to vote left or to vote right
27:42or to uh before
27:46the right to hold guns or not and so on
27:50and then of course the Crux of the model
27:53will be to introduce some social
27:54interaction
27:56and so the social interaction I'm going
27:58to write as
28:00an interaction between spins
28:02which is sum over J of j i j s j s t
28:08and in the context of what I'm going to
28:10speak about today I'm going to imagine
28:12that all these jij's are positive
28:16or zero
28:22so jij means that your the the the the
28:26network of the jij uh tells you who I is
28:32listening to in order to make his
28:34decision at the next time step
28:37so the J is such that jij is non-zero
28:40are there the neighbors influencing I
28:43and then depending on what these
28:45neighbors are doing then you tend to
28:47follow their advice or you tend to
28:50imitate what they've they've done okay
28:53and I'll show you later on the little
28:55funny video of um illustrating how
28:59strongly we are influenced by what other
29:02people do but I guess this is you know
29:05if you do a little bit of introspection
29:06you know that this is real that uh what
29:10people around you are doing is uh
29:13is is very efficiently affecting your
29:16own decision
29:18anyway so at this stage uh this is the
29:21model the model tells you that what
29:23you're deciding at the next time step
29:26is determined by a combination of three
29:28aspects
29:30one is public information
29:33so if I go back to my example of cell
29:36phone or technology then public
29:38information is for example uh the price
29:42of cell phones if the price of cell
29:45phones is the very high you're not very
29:48likely to buy the cell phone so if you
29:50imagine that SI equal plus one means
29:53that you buy a cell phone and H is very
29:56negative then there's a chance that you
29:58will not buy a cell phone because of
30:00prices can also be because of the level
30:02of technology so to go back to the 90s
30:05in the 90s cell phones were not working
30:09very well it was a a lot of
30:13um interferences and and network
30:16problems and so it was not you know it
30:19was we're not really excited by the idea
30:21and the salary of buying a cell phone
30:24then there's your own idiosyncratic
30:27preferences so if you're a geek or if
30:30you're prone to buy new technology
30:33there's a lot of people like this then
30:35your hi will be large positive if on the
30:39other hand you're like uh my grandmother
30:42say and then you will wait a long time
30:44before
30:46getting convinced or even never get a
30:49cell phone at all
30:51um so hi will be
30:56a random variable but random not in the
31:01sense of time dependent but random
31:04across the population
31:13and it's going to be distributed
31:15according to certain row of H
31:18and so this row of H
31:22I'm going to assume that it has a shape
31:25like this
31:27gaussian for example or unimodal
31:31if it's if it has more than one maximum
31:34then what I'm going to tell you today
31:36needs to be a little bit amended but I'm
31:39not going to uh use this today so
31:42unimodal
31:46and it's characterized by some
31:49with Sigma
31:53the mean is irrelevant
31:56because the the mean I can always shift
32:00to zero I'm going to assume that e of H
32:03is 0
32:04because the the mean of H I can always
32:09put into Capital H okay if h i as a
32:14non-zero mean I can subtract the mean
32:16and add it back to hft and it's going to
32:19be a kind of public information
32:21so this is not this is without loss of
32:25generality that I can impose that the
32:28mean of H is zero
32:30and finally
32:31um
32:33the imitation term the social pressure
32:35term
32:44is characterized by some js
32:47and so J gives you another scale in the
32:51problem if you want Sigma is one scale
32:53and this J is
32:56so I'm going to assume that that this
32:59gigs are all of other JS they exist
33:02uh then you have two scales in the
33:04problem one is a sigma and the other is
33:08J and these two scales will give you a
33:11sense of whether
33:12heterogeneity of the population is
33:15dominant or whether
33:17social pressure is dominant so we'll see
33:20that what matters is actually the ratio
33:23of these two scales J over Sigma but
33:27intuitively you can think that if Sigma
33:30is very large if people are very
33:32heterogeneous
33:34then social pressure will play less of a
33:36role because people are stubborn in a
33:39sense and that you know some age are
33:41very large and these people they never
33:44get influenced by others whereas if
33:47Sigma is very small so in the case of
33:49very homogeneous population
33:51then the social pressure term will play
33:54a crucial role so that's already at the
33:57level of you know just the thinking
33:59about the model
34:00what we
34:01can imagine will happen
34:06okay so now let me explain with a graph
34:10what's happening in this model
34:15and then uh explain to you how one gets
34:18these results in the special case of a
34:22mean field model where the jij is again
34:25will be
34:26simple enough for the calculations to be
34:29completed
34:44foreign
34:57as I said just draw a graph
35:00and then explain how you can actually
35:03compute what's going on so what I'm
35:05going to
35:07assume
35:09is that
35:11as a function of time
35:14h of T starts at minus infinity
35:17and ends at plus infinity
35:22so if you look at this equation here
35:25it means that whatever social pressure
35:28and idiosyncratic preferences when age
35:31goes to Infinity these two terms are
35:33negligible
35:34and therefore everybody
35:37takes the decision minus one okay
35:41so everybody is staying on the sidelines
35:44if you want and then as H increases we
35:47expect that more and more people will
35:49get convinced to buy a cell phone or to
35:52vote left or right or to buy the stock
35:55market whatever and you expect that SI
35:58will progressively flip from -1 to plus
36:01one and when H goes to Infinity you
36:04expect that everybody's convinced and so
36:06all the sis are equal to one so in order
36:09to describe the system
36:11I'm going to introduce with in physics
36:14is called the magnetization of the of
36:17the system which is just the average
36:20decision
36:21so it's 1 over n
36:23um over I of s i okay and of course M
36:28depends on T because esi's will depend
36:30on t
36:31so what do you expect to find so of
36:34course m
36:35is between
36:37plus one
36:40and minus one
36:43yeah
36:45and what I'm claiming is that depending
36:47on the ratio of J over Sigma
36:50so this is this is going to be the
36:53important
36:55parameter in the model
36:57then you can have two different types of
36:59the
37:00curves
37:02I'm going to use three colors I know
37:04that you don't see the red very well but
37:06I guess that's
37:07it's not going to be too difficult to
37:10see what I'm drawing but tell me if
37:12really you don't see the red
37:16so
37:22so the first case well I'm going to draw
37:25in blue
37:27is J over Sigma equals zero
37:30and in this case what you're going to
37:32see is a perfectly continuous curve
37:35so I'm drawing it like this
37:42and if J equals 0
37:44we're going to write an equation for
37:46that but essentially you're going to
37:48convince people one after after the
37:50other
37:51so
37:53um
37:54for age very negative
37:56in my example in my geek example the
37:59hi's that are very very positive will be
38:02able to flip fast and then as you
38:05increase age you're going to convince
38:07more and more people and then at the end
38:09of the process you'll convince the
38:12people who have dhis very negative so
38:16who are less prone to buy the new
38:18technology and so this is going to be a
38:21continuous curve which we'll see is
38:24going to be very simply related to the
38:26distribution of age it's actually the
38:28cumulative distribution of H that you're
38:31seeing here so if the distribution of H
38:33is smooth then it's cumulative is also
38:36smooth and you get uh this type of graph
38:39okay
38:41so this is a well-known problem in
38:44marketing uh and people have studied
38:48this in the marketing area is you know
38:53how to describe this curve how to model
38:56this car so this is usually called the
38:58early adopters
39:11and and this is the late adopters
39:19and in general this is an s-shaped curve
39:22that tells you how a given product a
39:26given new product
39:27is going to invade a market
39:30but now if you increase
39:34J over Sigma so
39:38I'm going to draw in in green the case
39:43where J over Sigma is small positive to
39:46the small then what you get is
39:48essentially the same kind of curve
39:50except that it's going to be
39:54steeper like this
39:58and the intuition is that at the
40:01beginning
40:02it's actually below the blue curve so my
40:05drawing is not very good already sorry
40:07for that it's below the green the the so
40:10look at this part it's below the the
40:12blue curve because because of social
40:15pressure which is not dominant but it it
40:17exists then because you see that other
40:21people are
40:23uh not buying they are not convinced you
40:27think well you know maybe uh maybe it's
40:29not a good idea to buy now uh and
40:32therefore you need better public
40:35information in order to be convinced but
40:37then as people get more and more
40:39convinced you see that there's a
40:41steepening of the curve which means that
40:43the speed of adoption is increased
40:45compared to the case J equals zero
40:47and and then you slip from one state
40:51minus one to the other state just one at
40:55at a more uh rapid rate okay
40:59and again you know thinking back uh
41:01about the 90s I remember very well that
41:04in the 90s at the beginning not only the
41:08technology was pretty bad but also
41:10prices were high but also many people
41:13considered completely ridiculous to work
41:16in the streets with a cell phone or to
41:19speak uh in a cell phone uh in public
41:23and so on so there was you know a real
41:26social pressure that prevented cell
41:29phones to invade more rapidly the market
41:34and then finally if J over Sigma is
41:38large which means J over Sigma
41:42larger than some critical value which
41:44I'm going to call AC
41:48and we'll see how this critical value is
41:51determined then you get a completely
41:54different curve and what you get is
41:57something that
41:58starts very low and stays very low
42:03because social pressure is so high that
42:07nobody actually flips so if you think of
42:11I don't know the tax evasion for example
42:14you know you can think that the the the
42:16the social pressure telling you well
42:19it's ridiculous to pay taxes you
42:21shouldn't pay taxes and so on it's so
42:23strong that people actually don't pay
42:25taxes until it becomes really costly to
42:29dodge taxes so that will be hft hft
42:32would be governmental measures to punish
42:35people who don't pay taxes in this case
42:37and so the level is very low
42:41and actually it stays low beyond the
42:43point but beyond the special Point here
42:46where uh the slope of the curve in the
42:49absence of
42:51interaction is maximum so you see this
42:54point here is the point where the blue
42:55line has the maximum slope is the point
42:58where the green line has a maximum slope
43:01and you have to wait
43:03uh beyond the time that you would have
43:06waited in the absence of social pressure
43:08to see something and what you see is is
43:12not an acceleration of this line here
43:16and then the sudden discontinuity
43:21foreign
43:38like this
43:43so this is a discontinuity with uh
43:46a jump of size that I'm going to call
43:48Delta
43:50and
43:51you see here really there's something
43:53completely different that happened which
43:56is that suddenly a finite fraction of
43:59the population flips more or less at the
44:02same time
44:03here on the other hand in the case When
44:06J is small is really individual flips
44:08that that build this curve this
44:11continuous curve in the large end limit
44:12but if you zoom you see that each time
44:15you see one or a few flips
44:18whereas in this particular case at one
44:20point there is in a sense the same
44:23effect as I was talking about in the
44:25context of the Galton Watson model and
44:28you'll see that the analogy is real
44:30there's there's an infinite size
44:33Avalanche of people flipping together
44:36so what it means here is that as people
44:39as more people flip
44:41they're going to convince more people to
44:43flip and so the question will be if I
44:47flip how many people
44:49do flip because of me
44:51and as you can imagine if the r0 is the
44:54propagation rate of this conviction
44:57mechanism is greater than one then some
45:01catastrophic events will happen and this
45:03is exactly what's going on in this model
45:05and that I'm going to show you uh in
45:08details in a few moments
45:12okay so first
45:15Innovation compared to the case without
45:18limitation
45:19with documentation curves are smooth
45:21with a strong enough imitation there's a
45:25sudden discontinuity that that happens
45:28now the second thing that is interesting
45:30about this model is what happens on the
45:33way back okay so imagine that of course
45:37in the case of cell phones this is silly
45:39because the technology never goes back
45:42in time but imagine that we're speaking
45:44about an economy or the stock market
45:48then you can think of this
45:52red curve as
45:55the number of people who are who are
45:57optimistic about the states of the
45:59economy
46:00and so in the case where they're strong
46:03enough limitation you're going to remain
46:05pessimistic because other people are
46:08pessimistic until a point where
46:10okay news are better hft has increased
46:13but also social pressure is such that
46:16suddenly everybody's extremely
46:18optimistic and starts buying houses and
46:22spending and so on and so you can have
46:24this boom phenomenon that sometimes
46:27happen in an economy
46:30so okay so that was on the way in so I'm
46:33going to add a little arrow here so
46:35that's what happened on the way in
46:38but then imagine that uh news get bad
46:42again
46:43you know for whatever reason
46:46um the objective news about the state of
46:48the economy degrades
46:50and therefore you're going to go back in
46:54regard so what happens in that case well
46:57in that case
46:58when h of T starts back from plus
47:02infinity and then decreases
47:04it's not the case that you're going to
47:07follow the same curve backwards actually
47:10you're going to follow
47:11the the equivalent of the lower branch
47:15for a while
47:18and then at this point here you're going
47:20to jump
47:22to
47:25the second equilibrium
47:29so there's a region here
47:32where what's called in physics the
47:34hysteresis Loop
47:36and which is a very interesting
47:39phenomenon
47:41and this history this Loop happens
47:43because for the same value of H
47:46there can be more than one equilibrium
47:48state so if I take this value of H for
47:51example
47:52then you see that there are
47:55two possible equilibrium states that are
47:58that will emerge from the from the
48:00equations and therefore which one you
48:03choose depends on history for example
48:06and if you if the history is such that
48:09you come from the lower Branch you will
48:11stick to the low Branch for a while
48:12before jumping to the second equilibrium
48:14and if you come from the high
48:16equilibrium Branch you pick on the high
48:19equilibrium Branch until you jump
48:22to the lower equilibrium branch and
48:25these points here are exactly the points
48:27where
48:28instead of having two equilibrium
48:30there's only one remaining
48:32so at this point you don't have a choice
48:35anymore at this point you had a choice
48:37the choice being a pessimistic or
48:40optimistic but at this point it's no
48:43longer an option uni point in which you
48:47can be is the optimism space state
48:51and here again below this value of H the
48:55only Equity room that survives is the
48:57low uh the pessimistic state that you
49:00want
49:01so this is really interesting because it
49:03shows that in some cases
49:06um you jump
49:07but it's not because the the other
49:11equilibrium state did not exist before
49:13it's just because you were stuck on the
49:16one that you started with okay
49:21so this is a very generic model for many
49:25things but in particular for the
49:26appearance of discontinuities in the
49:30evolution When the evolution of public
49:33information is smooth so you see this is
49:35really the main message is that in all
49:38the examples I mean in implicitly here
49:41I'm assuming that hft increases
49:44progressively or decreases progressively
49:47so public information is smooth and
49:50continuous
49:53smooth
49:55and continues
49:58or continuous and smooth if you prefer
50:01but the the reaction of the system
50:06m
50:07for one over n
50:09some of Y of Si
50:12is discontinuous can be discontinued
50:20and for me this is one of the big
50:21Paradox of the big mystery of social
50:25sciences or or stock markets is that
50:28indeed in many cases you see that public
50:31information is progressively unfolding
50:34but the reaction of the market is can be
50:38extremely violent and you know we're I
50:42don't know how many of you are following
50:43what's going on in the in the stock
50:46markets or in the bond markets actually
50:48right now but it it may be a case where
50:52there are two equilibria
50:55behind what's going on and we're on the
50:59verge of flipping from one to another I
51:01don't know of course but there are
51:03reasons to believe that such a scenario
51:05might be at play as we speak
51:10Okay so
51:13Let Me Now go to
51:16[Music]
51:17um
51:22telling you a little bit how do we get
51:25these results
51:27foreign
51:42limit
51:49but what's interesting is that the the
51:51phenomenology that I've drawn
51:55for you is actually much more General
51:57and if you even if you have
52:00a generic Network model describing
52:04social pressure then in a very large
52:07number of cases or very broad variety of
52:11networks uh the phenomenology that I'm
52:14going to talk about is valid I'm going
52:17to tell you at the end how some networks
52:19are different but you know it's the mean
52:23field
52:24um
52:25approximation is not inventing a
52:29phenomenology that doesn't exist in more
52:31realistic cases it's a pretty faithful
52:36uh
52:37approximation so what I'm going to
52:39assume is that jij
52:42is equal to j0 over n
52:45for all paths
52:48which means again that everybody is
52:50connected to everybody else
52:53and I'm going to give another
52:54interpretation that's a little more uh
52:57convincing in a second
52:59but for now you imagine that you take
53:03your cues from a very large number of
53:05people and all of them have the same
53:08influence on on you
53:11so in that case
53:15if I compute sum over J of jij
53:19SJ of t
53:22then if all jijs are equal this is equal
53:25to j0 over n
53:28um over J of s j f t
53:33but from my definition 1 over n sum of J
53:36of f s of J of T is equal to
53:41um M so this is j0
53:44times m C okay
53:49so here of course you know
53:52if I
53:54had to be pedantic I should exclude I
53:57from the sum or assume that j i i is
54:00zero
54:02and so of course you could you know
54:06quibble on the fact that there's the
54:08n minus one term here which is not
54:11exactly equal to M but
54:14what I'm going to say is of course
54:16always true in the large end limits so I
54:19can neglect the difference between n
54:21minus 1 and M
54:24okay so now
54:27what I can get is a very simple
54:30self-consistent relation that is going
54:33to tell me how M evolves because if I
54:36sum this equation
54:39uh on both sides so what I do here is I
54:43do 1 over n
54:45thumb over I of this 1 over n
54:50sum over I
54:53and what you get
54:56is that
55:00M of t plus 1
55:06is equal to
55:09um
55:091 over n
55:12the sum over I
55:14such that h i is greater than
55:18um
55:19minus h of t
55:22minus j0 M of t
55:27of plus one
55:32so you see this sign here is equal to
55:34plus one if h i is actually large it's a
55:37very large compared to minus H minus j0
55:41M of t plus
55:441 over n
55:47um over I such that h i is less than
55:50minus h of P
55:52minus j u m of t
55:55minus minus one
55:58okay
56:01so in order to give this equation uh a
56:05better looking shape
56:24foreign
56:36which can be which will be useful which
56:40is a probability
56:43well which is one over n
56:47sum over I of
56:50uh
56:51such that
56:53well
56:55such that s i is equal to plus one so
56:59it's the fraction
57:01not a probability is a fraction
57:05of spins
57:08equal to plus one
57:12and it's clearly related to m
57:18so m
57:21is equal to 2 Phi
57:23minus one
57:27so it's just a trivial transformation
57:30from M to Phi
57:33and then what you get is that
57:36Phi
57:39is given by
57:42what I'm noting
57:44key
57:46larger than
57:47minus H minus j0m
57:52which is key
57:56minus h plus j0
57:59minus two J zero Phi
58:03where this quantity here is just the
58:05cumulative distribution function so pH
58:08of x
58:10is the integral from X to Infinity
58:13pH of rho of H
58:18so row of H again is the distribution of
58:20idiosyncratic preferences
58:24and this is the cumulative or
58:27complementary cumulative distribution so
58:29it's the probability that small H is
58:31larger than some quantity and so you see
58:35that
58:36oops
58:37I've forgot the the times you see that I
58:41can convert this equation here
58:44into uh this equation here so what I'm
58:48actually Computing is 5 t plus one
58:52is equal to P of
58:56there's a t index everywhere
59:10Okay so
59:13so the trick the trick that helps you
59:16solving the model in the mean field case
59:18is that you can have M or Phi appearing
59:22in both sides of the equation which of
59:25course would not be the case if
59:27the jijs were only local because in this
59:31case what would happen is that there are
59:33only local magnetization local appeals
59:36and not the mean field that you can
59:39compute from the left hand side by
59:41summing it over I
59:44so of course this is a very special
59:46situation but as I said it's not going
59:48to be
59:50terribly different from a more realistic
59:53Network so why is that
59:56assumption not that irrealistic well you
59:59see that what it means if you see this
1:00:03equation here
1:00:05it means that social pressure
1:00:11is actually equal to
1:00:13or proportional to the fraction of
1:00:16people who have already adopted or the
1:00:18fraction of people who think left or
1:00:21think right or any other ways you want
1:00:24to
1:00:25picture the the model but this means
1:00:29that actually you're sensitive to Paul
1:00:32you're sensitive to surveys
1:00:34because surveys are are supposed to give
1:00:37you an indication of what the other
1:00:39people think
1:00:40and so in a sense it's not such a
1:00:43ridiculous assumption to use this mean
1:00:45field approximation because it means
1:00:47that what influences you is not your
1:00:50network of friends but it's
1:00:54um the opinion of the majority or at
1:00:57least
1:00:58a
1:01:00proxy for that which is given by
1:01:04um for example surveys or polls that are
1:01:07published in newspapers and in the case
1:01:09of stock markets you can think of the
1:01:13price the price of the stock market as a
1:01:16way to indicate how many people are
1:01:19optimistic or pessimistic
1:01:21so the fact that the the public
1:01:25information so to say is related to
1:01:28social pressure is not such a crazy idea
1:01:32after all
1:01:38Okay so
1:01:40foreign
1:01:43just erase these definitions and rewrite
1:01:46them to have a little more room
1:01:55so I've introduced two notations that
1:01:57are related M and Phi
1:02:01and I'm going to flip between them
1:02:03depending on uh the problem I'm
1:02:05discussing but let me skip let me step
1:02:08back to M and so what I got was
1:02:12in terms of M M of t plus 1
1:02:15is
1:02:182
1:02:19p
1:02:21the 2 minus 1 coming from the relation
1:02:23between Phi and M
1:02:25minus h of t
1:02:28minus J M of t
1:02:32minus 1.
1:02:36so this is what dynamically sets the
1:02:39evolution of the average opinion
1:02:42so let me
1:02:44try to see whether there are stationary
1:02:48solutions to this equation so imagine h
1:02:50of T stops changing with time
1:02:53and it's given the sun value h
1:02:56is there a value of M such that this
1:02:59equation is the stationary so I'm
1:03:02looking for six points I'm looking for
1:03:06possible solutions of the equation M
1:03:08Star equals
1:03:10to P larger than minus H which I'm
1:03:15assuming to be static for a second
1:03:18minus one
1:03:22so what are the solutions of that
1:03:29and you see this is really the question
1:03:31I need to answer to know whether I can
1:03:34have two possible points where I sit or
1:03:37only one
1:03:40okay well then as usual you know you
1:03:44need to solve such an equation and the
1:03:48way to solve this equation or to get
1:03:50some intuition is to make a drawing
1:03:54again so as a function of M
1:03:57let me plot the right hand side
1:04:00so
1:04:03again m is between -1 and plus one
1:04:08so I have the diagonal which is going to
1:04:11look like this
1:04:12and then depending on the shape of
1:04:16P larger than I can have curves that
1:04:20look like
1:04:22like this
1:04:25for example
1:04:28so in this case there would be only one
1:04:30solution
1:04:37but I can also have cases where
1:04:41this looks like
1:04:45like this
1:04:46and in this case I would have three
1:04:48solutions
1:04:59and in the case where there are three
1:05:00solutions actually you can look at the
1:05:03slope
1:05:04of these curves at the inspection point
1:05:07and you quickly realize that when there
1:05:09are three solutions only two of them are
1:05:11stable because they have slopes less
1:05:13than one and one of them this one is
1:05:17unstable because it has a slope larger
1:05:19than one
1:05:20and so in the case where there are three
1:05:22solutions to this equation there will be
1:05:24only two solutions that are stable and
1:05:27one solution that is unstable
1:05:30but in fact this solution exists the
1:05:33start solution exists so if I had to be
1:05:35slightly more
1:05:37besides
1:05:39on this graph
1:05:41I should actually add a dotted line
1:05:48that does like this
1:05:51which is the location of the third
1:05:52solution
1:05:55and although you know you cannot sit on
1:05:58this line so it's it's not a line that
1:06:00really exists
1:06:02um it has some important
1:06:05um I mean the existence and the location
1:06:07of this line as some important
1:06:09consequences
1:06:10in in the case where you start the
1:06:14system
1:06:15with a random
1:06:16uh Choice with with people in the random
1:06:20state
1:06:21and for a given value of H and then
1:06:23depending on where you start from so if
1:06:25the initial point is this point you're
1:06:27going to flow to this solution and if
1:06:31the initial point is here you're going
1:06:32to flow to that
1:06:34solution so if instead of having an age
1:06:38that very varies with time I have an age
1:06:40that's fixed in time but a population
1:06:43that starts randomly with a with the
1:06:47random fraction of people that are
1:06:48convinced who are convinced then
1:06:51depending on the position of this the
1:06:53initial condition compared to the
1:06:55unstable solution you go one way or the
1:06:58other so in some cases it's really
1:07:00important to know where this unstable
1:07:02solution lies
1:07:05so of course many of you will have
1:07:07recognized uh the story of the van der
1:07:10waals liquid gas transition and it's
1:07:13it's actually very very similar to the
1:07:15phenomenology except that here uh we are
1:07:18formally at zero temperature
1:07:20uh and the role of temperature is going
1:07:23to be uh what I'm going to talk about
1:07:25in the next lectures
1:07:29I mean the role of temperature of the
1:07:32analog of temperature in the social
1:07:33context
1:07:40so what determines whether one has one
1:07:43solution or three solutions well of
1:07:45course what determines the transition
1:07:48point
1:07:49is the case where
1:07:52one flips exactly from uh precisely at
1:07:56that point from one solution to three
1:07:58solutions so let me try to draw
1:08:01um
1:08:04in green
1:08:06what's going to happen so imagine that
1:08:08I'm pushing the Blue Line a little bit
1:08:10to the right
1:08:12and at one point
1:08:15I will have something like this
1:08:20okay so I can deform the Blue Line in
1:08:24such a way that these two solutions
1:08:26merge into a single solution and then
1:08:30there's a second one that is here okay
1:08:33so this is the critical situation that
1:08:37distinguishes the case where there's
1:08:39only one solution
1:08:41beyond that because you see beyond that
1:08:43I'm going to completely lose
1:08:48this solution here and just
1:08:50retain the the lower one but at this
1:08:54particular point I'm flipping from three
1:08:57solutions to two solutions and then to
1:08:59one solution so what determines the this
1:09:03critical case well it's two equations it
1:09:06will tend to see that that must be
1:09:08simultaneously obeyed one is that M Star
1:09:13still a based uh this equation so I
1:09:16still must have
1:09:18sort of critical point
1:09:26so the critical point is determined by
1:09:28The Joint solution of two equations one
1:09:31is the same as this one because you see
1:09:33this is the line that I'm uh drawing for
1:09:37the right hand side and it still has to
1:09:39touch
1:09:39uh x equal one line so I have to have
1:09:42that simultaneously M Star is two key
1:09:46greater than minus H minus J M Star
1:09:51minus one so I'm just repeating
1:09:54but the other Criterion is that at this
1:09:58point the slope of the green line must
1:10:01be equal to one okay
1:10:03you see from the graph that you lose the
1:10:06solution by having a point here that's
1:10:09tangent to the x equal y
1:10:13line
1:10:15and so in order to get the slope I need
1:10:19to take the derivative of the right hand
1:10:22side with respect to m
1:10:25so if I do that
1:10:27I will have that
1:10:30minus J
1:10:32times 2
1:10:35so a
1:10:41let me first notice that d p
1:10:46by the x is equal to minus rho of x
1:10:52okay
1:10:53trivially so if I take the derivative of
1:10:57this guy with respect to M I will pull
1:11:00out a minus J but the minus will go away
1:11:03with this minus here and so I'm going to
1:11:05have that the derivative is given by 2 K
1:11:10rho
1:11:12of minus h
1:11:15minus J
1:11:17and star
1:11:21um
1:11:21that must be equal to one
1:11:26okay
1:11:28so what does that mean it means that if
1:11:31I'm give if I give myself a value of J
1:11:35so if J is sixth then these are two
1:11:38equations determining M star and H
1:11:42and this will be the critical value of H
1:11:44Beyond which there is no longer three
1:11:47solutions but one
1:11:49so these two equations seen as an
1:11:52equation determining H and M star for a
1:11:56given J
1:11:57these two equations are the ones that
1:11:59determine
1:12:03these two points
1:12:05okay
1:12:07these two points here I'll determined by
1:12:09the last point where three solutions
1:12:11exist
1:12:17s
1:12:22of course you can think of it
1:12:24differently you can think of the problem
1:12:27at a given value of H
1:12:30say this one
1:12:33and then these two equations would
1:12:35determine M Star NJ and J would be the
1:12:41first value or the the critical value
1:12:44for for this value of H for for a given
1:12:46value of H the the the the amount of
1:12:50imitation that you need to include in
1:12:53order to get three solutions so
1:12:55depending on the way you want to think
1:12:56about it uh either at 6h letting J vary
1:13:00or at 6j letting H vary you get uh the
1:13:05value that you need so in particular in
1:13:08the graph here I was assuming that J is
1:13:10Sixth and that I'm varying H and this
1:13:13point here is the point where uh
1:13:17the the three solutions
1:13:20merge into two and then become one
1:13:24so what am I why am I insisting on on
1:13:27this Criterion
1:13:28well I'm insisting on this Criterion
1:13:30because I want to give you another
1:13:31interpretation of this Criterion which I
1:13:34think is the very physical or
1:13:38very intuitive in terms of what what
1:13:41goes on in the system
1:13:42so again what I said was that in here
1:13:46what what happens when H increases is
1:13:49that progressively you flip more and
1:13:51more people from the pessimistic state
1:13:53to the optimistic state
1:13:55but these people by flipping
1:13:58they encourage other people to flip as
1:14:00well and the question is whether this is
1:14:03going to
1:14:04be limited to small clusters or is
1:14:08actually going to give rise to a
1:14:11a full Avalanche
1:14:13so
1:14:18let me explain to you why this the
1:14:21Criterion here is actually exactly
1:14:23telling you that there's another large
1:14:24of infinite size that happens in the
1:14:27system
1:14:28foreign
1:14:34so again let me write the equation that
1:14:37I need f i of t plus 1 is the sine
1:14:42of H
1:14:45plus h i
1:14:49Plus in the case of mean field
1:14:53well in the general case sum over
1:14:57j0
1:14:59and
1:15:03just can you still see what I'm writing
1:15:13yes
1:15:16but I shouldn't go
1:15:18much farther
1:15:23so imagine that one of this of the spin
1:15:25flips
1:15:34from
1:15:35-1
1:15:37two plus one
1:15:40okay
1:15:42then what other people are going to see
1:15:46is that
1:15:48the social pressure that they had
1:15:50previously j0 times m
1:15:55is increased it's increased by a certain
1:15:58quantity which is j0 times M plus
1:16:022 J over n
1:16:07because suddenly you know if you zoom
1:16:09into the terms contributing to j0 times
1:16:12m
1:16:13let me write again what this j0 m means
1:16:16it means j0 over n
1:16:20sum over J of f j f t
1:16:25so if one of these guys changes from -1
1:16:29to plus one then the mean field seen by
1:16:33others changes from j0m to j0 M plus 2
1:16:38over n 2 because s is gone from minus
1:16:41one to one so there's a jump of Two And
1:16:43it contributes by j0 over n
1:16:46to the mean field okay
1:16:49but because this jump has happened
1:16:53other sites which were previously
1:16:55convinced to be down will have a
1:16:59tendency to flip upwards and what it
1:17:02means is that
1:17:04now if I take a certain
1:17:06site l
1:17:08some agent l
1:17:10this agent L can be such that hft
1:17:15plus h of L
1:17:18Plus j0m
1:17:21was negative so it was happily sitting
1:17:25in a pessimistic state but
1:17:30suddenly h of t
1:17:33plus h of L
1:17:35Plus j0m
1:17:37plus to J zero
1:17:40Over N is positive
1:17:43okay
1:17:45and so if this happens the fact that
1:17:49a given the spin has flipped a given
1:17:52agent has changed from -1 to 1 is going
1:17:55to induce the flip of a Second Spin
1:17:57which is H of L and this can carry on
1:18:00for a while until the Avalanche either
1:18:04uh stops or grows forever
1:18:07and so what we know is that we need to
1:18:10compute the probability for this to
1:18:11happen that is the probability for one
1:18:14spin flipping generating a Second Spin
1:18:17to flip and if this probability is
1:18:20greater than one it's going to explode
1:18:22and if this probability is less than one
1:18:23it's going to stop
1:18:26so what I have to do the question that I
1:18:29have to answer is simply what is the
1:18:32probability that a given spin is in a
1:18:35situation or a given agent in a
1:18:37situation where these two conditions are
1:18:40similar simultaneously
1:18:42um obeyed
1:18:44well it's it's uh
1:18:47simple to see that
1:18:56foreign
1:19:07I can regroup these two equations
1:19:11as the following HL must be between
1:19:15h
1:19:18plus minus sorry minus h
1:19:22minus j0
1:19:24m
1:19:26and
1:19:27minus h
1:19:29minus j0m
1:19:31minus 2j0 over n
1:19:37so what is the probability that this
1:19:39happens
1:19:40well it happens with probability
1:19:46uh the density of H
1:19:52minus J zero m
1:19:55times the width of this interval which
1:19:57is 2j0
1:19:59over n
1:20:01so the probability that a given agent L
1:20:05has a it's idiosyncratic field between
1:20:08these two numbers is given by the
1:20:11density distribution
1:20:12computed at one of the bounds times the
1:20:15width
1:20:16of the bound of the interval so if you
1:20:19want this is DH
1:20:22so that's the probability that one of
1:20:24them is in this situation and the
1:20:27probability that any H any L
1:20:30is prone to flip
1:20:32is n times that total probability
1:20:40which I'm going to call r0 because it's
1:20:43the priority that given that a given
1:20:45that a certain spin flips another spin
1:20:48what whoever it is Will spin will flip
1:20:51as well then the total probability is
1:20:54this result times the number of
1:20:57potential
1:20:58agents that can't flip which is n itself
1:21:02so you see that it cancels the factor 1
1:21:05over n and what we get is
1:21:12your Autos group
1:21:15I'm out of the screen yes sorry
1:21:21so the total probability so
1:21:22independently of who L is as I was
1:21:25saying is what I'm calling r0 this n
1:21:29times 2j0 over n times rho of minus H
1:21:34minus j0 m
1:21:38and so you see that all 0 equal 1
1:21:43is equivalent to the second condition
1:21:45that defines my critical point is 2j0
1:21:49sorry
1:21:51you should have told me before but I I'm
1:21:52missing a j0 everywhere
1:21:57uh 2J hero row of minus H star I mean
1:22:03this is the
1:22:06the points where I'm that I'm looking
1:22:08for
1:22:09but you see that the second condition
1:22:11just means that
1:22:14the epidemic or the the the way of being
1:22:19able to convince people
1:22:21the strength of conviction
1:22:23is marginal if it's lower than that
1:22:26equilibrium is stable and nothing
1:22:29happens one guy flips and maybe a few
1:22:32other guys are flipping but the
1:22:34Avalanche is soon stopping
1:22:36exactly as a in a sand pile where one
1:22:39grain dislodges the sun number of grains
1:22:41and then you have a large Subs but then
1:22:43if you're exactly at this critical point
1:22:47uh the Avalanche can grow very large
1:22:49it's eventually going to stop but if
1:22:52you're only slightly beyond that point
1:22:54then the Avalanche is going to invade
1:22:57the whole system and that's why
1:23:00you have this jump
1:23:02the jump
1:23:03corresponds to
1:23:06another launch invading the whole system
1:23:09but if you zoom close to this point here
1:23:13then all the funny statistics I told you
1:23:16about in the Galvin Watson model in
1:23:18particular you remember the the family
1:23:21size distribution and all these things
1:23:23then you can observe all these uh
1:23:27interesting properties when you approach
1:23:29this particular point and if you zoom in
1:23:31and instead of looking at this
1:23:33continuous curve you're actually think
1:23:35in terms of Agents flipping
1:23:39so the reason I'm telling you all this
1:23:41is is that actually these are things
1:23:43that you can measure in actual magnets
1:23:45it's much more difficult to see in the
1:23:48in in human population although there
1:23:52are things that you can measure as well
1:23:54but maybe not at this level of precision
1:23:56but in the case of magnets you can have
1:23:59actually a pretty good measure of these
1:24:02Avalanche sizes
1:24:05and and compare them with the theory so
1:24:09if you want to compare with Theory then
1:24:11the mean field approximation may be a
1:24:14good approximation to give you what's
1:24:16going on so the phase diagram if you
1:24:18want this this opening of a hysteresis
1:24:21Loop but it might not be good enough to
1:24:23explain the exponents for example the
1:24:26the actual distribution of family sizes
1:24:30is not correctly described by
1:24:33um
1:24:33by the the mean field approximation if
1:24:36you want to actually compare to
1:24:39three-dimensional real magnets but this
1:24:42is the
1:24:43this is maybe a little bit of a detail
1:24:48okay so uh before making a pause let me
1:24:54um
1:24:55finish by two remarks one is that
1:25:00as I told you if for example you're not
1:25:04in the Midfield limit but you're uh as I
1:25:07just said on the D dimensional graph
1:25:11so it's your spin this time or if your
1:25:14agents are on a regular graph like this
1:25:19or on a tree with a finite number of
1:25:23Neighbors
1:25:24or for example in this case
1:25:30then
1:25:31except for the the detailed match nature
1:25:34of the exponents that I just talked
1:25:36about
1:25:37um
1:25:38the physics is the same you you have the
1:25:40opening of a hysteresis Loop if the
1:25:43dimension of space or if the number of
1:25:45neighbors is large enough
1:25:48but um
1:25:51but it disappears this transition
1:25:53disappears when uh
1:25:56when the dimension or the number of
1:25:58neighbors is not large enough so for
1:26:01example in this case as soon as the
1:26:03dimension is greater than two
1:26:06then AC is finite
1:26:10and in this case as soon as the number
1:26:12of Neighbors
1:26:13so here I'm assuming that it's a regular
1:26:16graph so all my all Sites have the same
1:26:18number of neighbors as soon as it's
1:26:21greater or equal to four
1:26:24what I've just told you is the is
1:26:26correct there is a critical value AC
1:26:29but if the graph is a small enough
1:26:31Dimension or if there are not enough
1:26:34neighbors then AC goes to Infinity
1:26:37there's no longer any discontinuity that
1:26:39that occurs
1:26:42so that was the first remark
1:27:05three marks
1:27:07so the first one I just did V greater
1:27:10equals to okay greater or equal to four
1:27:15the second remark is
1:27:18the behavior of Delta
1:27:20Delta remember is the uh
1:27:24is the amplitude of the jump
1:27:27so you can draw Delta as a function of
1:27:32say over Sigma
1:27:36and if Jo Sigma is less than AC
1:27:40and of course Delta is zero
1:27:43because it's the continuous evolution
1:27:48but if J over Sigma is greater than AC
1:27:51then from what I said it's the point
1:27:53where
1:27:54you know you have this this Regis Loop
1:27:56opening and then there's a critical
1:27:59growth
1:28:00of Delta as a function of J of J C it
1:28:03goes like this
1:28:06goes to 2.
1:28:11and
1:28:12what happens here
1:28:14well depends on the lattice
1:28:18it depends on the dimension of space but
1:28:20in mean field
1:28:23is the square root so it's the square
1:28:25root of J over Sigma minus 80.
1:28:30but as I told you in real
1:28:32three-dimensional magnets for example
1:28:34this would not be a square root but a
1:28:36slightly different Power
1:28:38the third actually close to the third
1:28:40but the idea here is that there's a
1:28:44continuous growth of Delta as J over
1:28:47Sigma becomes larger and larger
1:28:52and the third remark is that actually
1:28:55this random field icing model
1:28:57has a long history also in the social
1:29:00sciences and in economics
1:29:03where it's called the
1:29:05I mean a special case of this model is
1:29:09called the selling
1:29:12okay
1:29:13chronovata
1:29:19model
1:29:22so
1:29:24chatting with an economist granavatar is
1:29:26a sociologist and they wanted to
1:29:29understand for example how riots emerge
1:29:33and continue or have a seminar in in the
1:29:38sense of
1:29:40a lecture happening every week is either
1:29:43losing its audience or actually gaining
1:29:47an audience and stabilizing and so what
1:29:51these guys have in mind is that the
1:29:54fraction of attendees
1:29:57to a seminar or to a riot
1:30:01is given by something like
1:30:04integral from 1 minus 5T
1:30:07to Infinity
1:30:10of
1:30:11something that in order to be
1:30:14close to what I told you could be
1:30:16written like this row of H pH so what
1:30:19they have in mind is that people have a
1:30:22intrinsic propensity to join a riot or
1:30:26to follow a lecture but if the number of
1:30:30people who attended the riot at the last
1:30:33rounds or attended the lecture at the
1:30:36last round is large enough then even
1:30:39people who are not very convinced will
1:30:41join so it's really the same idea as the
1:30:44random keyalizing model there's a social
1:30:46pressure which is in which is measured
1:30:49by the the fraction of the population
1:30:51that joins the riot or the fraction of
1:30:54the population that joins the seminar of
1:30:56course a fraction of the target
1:30:58population I don't expect the whole of
1:31:00France to come follow my lectures but um
1:31:03you see what I mean so if you have an
1:31:06evolution like this then you can see uh
1:31:10very easily that this corresponds to the
1:31:12random feeling model and mean field
1:31:15with
1:31:18uh j0
1:31:21equal h
1:31:22equal one-half
1:31:26and depending on the shape of row of H
1:31:30you can either have
1:31:32three solutions or one solution
1:31:37and so the evolution of the
1:31:41of the seminar actually resembles very
1:31:44much what I was uh talking about here so
1:31:47the idea of selling in Grana Vetter is
1:31:49that if initially the attendance is
1:31:52large enough
1:31:53then the the riots or the seminar will
1:31:56uh
1:31:58will prosper and grow but if you on the
1:32:01other hand the number of people
1:32:02initially joining the riot is is too
1:32:04small then it's going to picture out
1:32:08and the points where
1:32:12things changes
1:32:14this unstable solution of the equations
1:32:17that I talked about
1:32:19this guy here
1:32:21acts as a separate tricks between the
1:32:23two behavior and that's what what these
1:32:26people call the Tipping Point
1:32:34so you see it's very important as a as a
1:32:36concept it's a Tipping Point between uh
1:32:39social unrest actually gaining the whole
1:32:43population or actually featuring up
1:32:46without doing much just through social
1:32:49pressure so if you're just below the
1:32:51Tipping Point things are going to get
1:32:53better if in the case of riots not in
1:32:56the case of a lecture where people will
1:32:58finally
1:32:59leave completely the lecture exception
1:33:03aficionados who are the guys
1:33:06contributing to this point but if you're
1:33:09only slightly above this Tipping Point
1:33:11then things will go bad okay
1:33:14so this is to put in context of a
1:33:17of a long history of
1:33:19of something that has never been called
1:33:22the random field icing model and
1:33:24actually is a is a special case of it
1:33:26but that has a long history in the
1:33:29social and economic literature so let me
1:33:33stop here uh and take a pulse for 15
1:33:37minutes but before stopping I want to
1:33:39show you a funny little
1:33:41movie
1:33:43um
1:33:44trick to illustrate the strength of
1:33:47social pressure
1:33:49so wait I'm going to share my screen
1:33:54um
1:33:59so I'm cutting the camera sharing the
1:34:01screen
1:34:07can you see the screen
1:34:10yes
1:34:12okay
1:34:17so
1:36:02um so of course this is a little bit of
1:36:05a joke but it's it's actually to
1:36:06illustrate how uh social animals we are
1:36:10and how strongly we're influenced by
1:36:12people so there's a lot of the real
1:36:15experiments to show this how we get
1:36:17influenced by what other people do and
1:36:19say but as you see as I try to
1:36:22illustrate in the models this can lead
1:36:25you know on large length scales on for
1:36:27large Aggregates to pretty spectacular
1:36:30effects that can you know be either
1:36:33detrimental or actually in some cases
1:36:37again thinking of vaccination campaigns
1:36:40maybe if there's a strong social
1:36:41pressure that allows people to get
1:36:44convinced that getting vaccinated is a
1:36:46good idea then it can actually promote
1:36:49promote
1:36:50the vaccination rather than oops
1:36:58than being detrimental so
1:37:03sorry
1:37:06and equip this and stop sharing my
1:37:09screen
1:37:18okay I can't go back to
1:37:22but I need so I propose to pause anyway
1:37:24and reconvene at 11.
1:37:27for the second half of the session thank
1:37:30you
1:37:50this conference will now be recorded
1:37:54there we go
1:38:01so I told you about the random view
1:38:02icing model in general now I want to
1:38:06retail the same story essentially but on
1:38:10a concrete example and emphasizing
1:38:13something uh interesting that will come
1:38:16out uh that's new with respect with
1:38:19respect to what I said but closely
1:38:21related so that's what I I'm calling
1:38:23Cliff Edge optimization you'll see why
1:38:35and I'm going to add a descriptor to
1:38:37this section which is different Cliff
1:38:40Edge optimization and the restaurant
1:38:41problem
1:38:49foreign
1:38:55problem because I I want to instead of
1:38:58speaking about these models in abstract
1:39:01so I want to give some flesh to
1:39:07to the story
1:39:08Okay so
1:39:09I'm going to uh assume that
1:39:14there is a restaurant which you know
1:39:16maybe just started and um and the
1:39:20restaurant owner will have to think
1:39:22about uh pricing his menu
1:39:24having in mind or maybe forgetting that
1:39:27social effects are very important for
1:39:30the better or for the worst I mean
1:39:32social effects means that if the
1:39:35restaurant is perceived as trendy then
1:39:38people will go there even if it's not
1:39:41that great and expensive
1:39:45um but maybe also there will be you know
1:39:48kind of catastrophes where suddenly
1:39:50people stop going to that restaurant uh
1:39:53because of social effects as well so I'm
1:39:56going to call Phi
1:39:58the occupation rate of the restaurant uh
1:40:02very similar to what I call Phi before
1:40:04you remember fire was
1:40:07um
1:40:08related to M so Phi equals zero means
1:40:10that nobody goes to that restaurant and
1:40:13Phi equal 1 is the full occupancy so
1:40:17it belongs to zero one
1:40:21from uh
1:40:23for occupancy
1:40:29to vacant
1:40:33and what determines why
1:40:36is a combination of
1:40:41idiosyncratic preferences people like
1:40:44some kind of food all their don'ts
1:40:46prices
1:40:48and
1:40:49um
1:40:50and social pressure so what I'm going to
1:40:53say is that Phi is the probability
1:40:56that
1:40:58h i the same interpretation as before
1:41:03which is called in this context the
1:41:05propensity or the willingness to pay
1:41:10often this is called the willingness
1:41:15pay
1:41:18so the probability that h i is greater
1:41:21or equal than
1:41:23the price minus uh
1:41:272 j0 I
1:41:33so it's really rephrasing in a slightly
1:41:35different context where p is what I
1:41:38called H before but now I want to think
1:41:40of it as a price directly and so what
1:41:43I'm saying is that you go to the
1:41:45restaurant if the price is sufficiently
1:41:47low or even if the price is high if
1:41:50there are enough people that who go
1:41:52there because the the C value of going
1:41:55to the restaurant even if it's a high
1:41:57price is is lower okay
1:42:01and in order to make things concrete and
1:42:04to compute things that I just alluded to
1:42:06and Drew uh graphs of General value
1:42:10before I'm going to assume that the rho
1:42:13of H
1:42:15uh so the willingness to pay is uh
1:42:20is gamma exponential of minus gamma h
1:42:24uh when h
1:42:27is positive and zero elsewhere
1:42:32so everybody is willing to pay a little
1:42:34bit to go to the restaurant
1:42:36but
1:42:38um
1:42:38but then as age increases there's an
1:42:42externality exponential decay of the
1:42:45willingness to pay uh to go to to the
1:42:49restaurant
1:42:50so in particular you remember something
1:42:53that
1:42:54is useful because it actually happens
1:42:58here already is the cumulative
1:43:01distribution which in this case is very
1:43:03simple it's explanation of mine gamma h
1:43:06for H positive
1:43:12okay so injecting this special shape in
1:43:16the general equation we find that Phi is
1:43:20equal to exponential of gamma
1:43:23the min
1:43:27JP yes
1:43:29and thanks for writing out of the frame
1:43:31of the video right now just for me maybe
1:43:34you can consume the camera a little bit
1:43:39okay
1:43:49okay thank you very much no no I'm sorry
1:43:56um two JP 2j5
1:44:01minus p
1:44:04and zero
1:44:07okay
1:44:12um
1:44:13so that's just injecting
1:44:16this shape into this equation
1:44:19and now I want to uh see what this means
1:44:23in terms of the solution
1:44:26so again the best is to draw a little
1:44:31graph
1:44:32so first thing I'm going to assume that
1:44:35p
1:44:37is greater than
1:44:392J
1:44:41in the first graph that I'm going to
1:44:43draw here and then I'll use another
1:44:46craft to a troll what's going on in the
1:44:49in the in the other case
1:44:51so as a function of Phi
1:44:54so Phi has a maximum value which is one
1:44:58and uh of course as usual one has to
1:45:03find
1:45:05intercept with this line here
1:45:08and if p is great greater than 2 J
1:45:12it means that actually
1:45:16um
1:45:17uh the curve looks like
1:45:20like this
1:45:25so this is exponential of minus gamma
1:45:28key
1:45:31and the point where
1:45:34the function reaches one
1:45:37is for uh Phi equals p over 2J
1:45:45okay
1:45:47but because I'm assuming that t is
1:45:49greater than to J this in section this
1:45:52this point here is beyond one so the
1:45:55only uh solution in that case is
1:46:00is this point here so this is this is a
1:46:03solution that I'm looking for five star
1:46:05and it's Unique
1:46:08okay
1:46:14now what happens if p
1:46:20if 2J is greater than p
1:46:24so if we know from the previous
1:46:27discussion that when social pressure is
1:46:29strong enough interesting things can
1:46:31happen
1:46:32well then again
1:46:37drawing this little thing here
1:46:39now the point where
1:46:42this function reaches 1 which is p
1:46:46equals to J Phi gives a solution which
1:46:49is below one
1:46:50but
1:46:51it's right here so five star
1:46:57Phi equals the two
1:47:00T over 2J is below one
1:47:04this is one
1:47:07okay and now again two things can happen
1:47:13let me draw it uh in two different
1:47:15colors one is that the curve does like
1:47:18this
1:47:21and of course after that it sticks to
1:47:24one
1:47:25okay
1:47:27so again this is explanation of minus
1:47:30gamma p
1:47:31and so you see that unit solution
1:47:33there's a again only one solution which
1:47:37is Phi equal one
1:47:45but there's another possibility which if
1:47:48p is larger than
1:47:50if if T is still larger so X natural
1:47:54minus gamma P starts lower and then you
1:47:57could have something like this
1:48:01okay and then in this case you see that
1:48:04there are three solutions
1:48:07sorry three solutions one two and three
1:48:11this is just a break point this is a a
1:48:14the point where the Min here switches
1:48:18from a non-trivial uh Evolution to
1:48:22um
1:48:23to zero and so we're five six to one
1:48:27so in this case you could have a a
1:48:30situation where either the attendance is
1:48:33full or it's uh it's actually close to
1:48:37vacant
1:48:38and of course as we saw in the previous
1:48:41case there's an intermediate physical
1:48:44case
1:48:45where
1:48:48something like this happens
1:48:51and where the intersection point
1:48:53also has a slope equal to one
1:48:56and so this the fact that the slope is
1:48:59equal to one is a signal that this
1:49:01solution is about to disappear
1:49:04or about to appear depending on which
1:49:06direction you go and as I explained in
1:49:09the previous lecture it's also
1:49:12associated with this our zero value
1:49:15which is the propagation of uh of of
1:49:18influence that uh was discussed in the
1:49:2315 minutes ago
1:49:25okay so let's write the critical
1:49:28conditions explicitly in that case
1:49:31so the critical conditions are fast that
1:49:355 star
1:49:36is a solution of this equation but a
1:49:39non-trivial one so we should keep 2J Phi
1:49:42minus P instead of 0 because it's
1:49:44interior to the to the domain so it's
1:49:48exponential of gamma
1:49:512J 5 star
1:49:55minus p
1:49:57okay and the second solution is that the
1:50:01slope of this function at this point is
1:50:03equal to one and the slope of the
1:50:05function is
1:50:072 gamma
1:50:08J
1:50:10exponential of minus of gamma to J star
1:50:15minus p
1:50:17to JC 5 Star minus p
1:50:20is equal to 1.
1:50:22okay
1:50:28so this is these are the same equations
1:50:31as I wrote in a general case before but
1:50:34now I'm more explicit because I I've
1:50:38given this a special shape to row of H
1:50:43and so you see for example what I can do
1:50:46is to
1:50:48um
1:50:51uh
1:50:52to use this equation here
1:50:56to extract the fact that the exponential
1:50:59function here is equal to 1 over 2 gamma
1:51:01J
1:51:03and so the first equation is going to
1:51:05give me something like
1:51:072 gamma J
1:51:10I star
1:51:11equal one
1:51:14okay
1:51:16but if I want a five star that is less
1:51:20than one so I want a solution that's
1:51:22interior to The Domain you see that the
1:51:25only possibility is that that Phi star
1:51:28is less than one is when 2J gamma is
1:51:32greater than one
1:51:36so that's the condition for uh for the
1:51:39existence of non-trivial effects
1:51:43and now
1:51:46um using
1:51:48this equation here
1:51:50and injecting it
1:51:52in
1:51:53in one of the two equations anyway
1:51:55they're going to be giving the same
1:51:59result I also find that two gamma
1:52:03J
1:52:04is exponential of gamma P minus one
1:52:11so uh gamma p
1:52:15is equal to log to 1 plus log
1:52:21of 2. gamma J
1:52:26okay so this defines a critical point
1:52:29for p the critical value for p
1:52:31or let me call it PC
1:52:37okay so with all this in hand I can now
1:52:40uh plot what's going on for a Phi Phi
1:52:45star itself
1:52:46so let me start by the simple case
1:52:50the simple case is when this condition
1:52:52is not met so when two gamma J less than
1:52:56one
1:52:57then you see that there's no possibility
1:52:59of satisfying these conditions
1:53:03so there's nothing uh non-trivial going
1:53:06on there's never any possibility of
1:53:08having more than one solution
1:53:10so this means in layman term that if
1:53:14social pressure is small enough then uh
1:53:17nothing special will happen for each
1:53:20price there will be an attendance so for
1:53:23each price there will be a demand for
1:53:26the for the good and there's it's a
1:53:28unique relationship between prices and
1:53:30demand
1:53:31so there's a well-defined demand curve
1:53:33and what it looks like
1:53:37as a function of gamma P this demand
1:53:41curve is very simple so I'm I'm drawing
1:53:445 Star the unique five star
1:53:47it's equal to one
1:53:51if gamma p is small enough
1:53:54up to
1:53:562 gamma J
1:54:00and Gamma key is equal to 2 gamma J then
1:54:02the curve starts going down
1:54:06because I'm logged
1:54:08so this is okay it's the it has a a
1:54:13an angular point
1:54:15but apart from that this is a you know
1:54:18standard uh demand curve when prices
1:54:22increase
1:54:23the demand goes down except then that
1:54:26when prices are lower than this social
1:54:30pressure term then everybody happy to
1:54:33pay whatever price and the restaurant is
1:54:36full okay
1:54:39so more interestingly what happens if
1:54:42two gamma J is greater than one
1:54:47then we know that
1:54:49some non-trivial solutions can happen
1:54:53and remember something that I've assumed
1:54:57here
1:55:00in order to be in that situation I've
1:55:02also assumed that P
1:55:05oh plus P less than 2J
1:55:09to remember that as well
1:55:11and so now what it looks like
1:55:16is closer to what we had in the generic
1:55:20random heelizing model
1:55:22so again Five Star as a function of
1:55:25price
1:55:27and now what you find is that actually
1:55:29five star
1:55:31remains stuck to one until
1:55:40two
1:55:42gamma J
1:55:45which is the
1:55:47equivalent to the fact that P must be
1:55:50less than 2J
1:55:51so it's equal to 1 all the way
1:55:56so that's the analog of this plateau
1:56:00and then here it jumps to
1:56:06a curve that goes down like this
1:56:09but now I have the same issue as before
1:56:12I have a history read this and the
1:56:15hysteresis curve can be computed and it
1:56:17looks like this
1:56:22and this point here is the point given
1:56:26by this equation so it's it's one
1:56:30plus log
1:56:33of
1:56:342 gamma J
1:56:38okay
1:56:40so that's the demand curve and you see
1:56:43that now because of social pressure
1:56:45there's this very strange phenomenon
1:56:47that is not standard in economics
1:56:50textbook where for the same price you
1:56:54can have two possible demands
1:56:58so if price is very low there's a unique
1:57:01solution 5 Star equal one
1:57:04so if the restaurant is both popular and
1:57:08cheap of course it's going to be full
1:57:11there's a also a branch here where the
1:57:15price is really high and in this case
1:57:17the whatever the the social pressure uh
1:57:21if the price is beyond some threshold
1:57:24people will stop going there
1:57:27and then there's an intermediate phase
1:57:30where for the same price you can have
1:57:32either your restaurant full or your
1:57:35restaurant you know half empty
1:57:38and so again what happens depends on uh
1:57:43history
1:57:45so in this multiple equilibrium cases
1:57:48there's history dependence the series is
1:57:51and so if you start if you're a
1:57:54restaurant owner and you start by low
1:57:57prices then you're going to fill your
1:57:59restaurant and because it's full
1:58:02people will like going there and so
1:58:05you'll remain full even if it's clearly
1:58:08overpriced but then you see at this
1:58:10point
1:58:11there's a jump of attendance and
1:58:14suddenly
1:58:15you know people realize that they've
1:58:17been going to this expensive restaurant
1:58:19not because it's especially good but
1:58:21mostly because people
1:58:24um have been going there and you've been
1:58:26uh you know following the crowd but
1:58:28suddenly price becomes such a an issue
1:58:31that you stop going there and because
1:58:32you stop going there other people go
1:58:34stop going there as well and there's
1:58:37another launch of uh departures which
1:58:40leads the restaurant owner with a very
1:58:43low attendance
1:58:44so you know you can imagine that and I'm
1:58:48going to go back to that in a second but
1:58:49you can imagine that the restaurant
1:58:50owner having realized that these price
1:58:53is now too high is trying to reverse
1:58:55courses of course and back pedals and
1:58:59now lower its price
1:59:01but unfortunately for him
1:59:03instead of uh recouping full attendance
1:59:07at the point where he lost it
1:59:10he's going to lower his price much lower
1:59:12he's going to have to lower his price
1:59:14much more in order to again jump to the
1:59:18High attendance Branch okay
1:59:20so that's what's going on
1:59:24but so what is interesting about this
1:59:26problem is that in traditional economics
1:59:30um
1:59:30agents optimize their utility of their
1:59:34households and their profits if they are
1:59:38firm
1:59:39so the restaurant owner
1:59:44is supposed to optimize his profit
1:59:49optimizes
1:59:53is or higher profits so it's
1:59:57process
2:00:01so what is the profit of the restaurant
2:00:03owner
2:00:05p
2:00:06it will depend on price
2:00:09and it's going to be given by
2:00:12uh the attendance that depends on price
2:00:16times the price minus
2:00:20production price so produce a menu he
2:00:24has to pay a certain amount P0 per
2:00:27client and if he charges P then the
2:00:31profit he makes is Phi or she makes is 5
2:00:34p 0 minus P okay
2:00:36and so now you have to optimize
2:00:39this profit to fix the price this is
2:00:43what is going to give you the price at
2:00:46which you should uh uh open your
2:00:50restaurant
2:00:52and in this case
2:00:53well because Phi is a decreasing
2:00:55function
2:00:57there's a there's a well-defined maximum
2:00:59and the the maximum of the price is
2:01:03somewhere
2:01:04uh here maybe
2:01:06so gamma P star
2:01:09and so if I plot the profit as a
2:01:11function of
2:01:14of p uh the profit in red
2:01:18will have a shape that's actually uh
2:01:21growing linearly uh here
2:01:27and then it's going to do something like
2:01:29this
2:01:31and that's the optimal process
2:01:35P Optimum
2:01:37foreign
2:01:43and if you change a little bit J so of
2:01:47course people don't know how much Social
2:01:50pressure people other people are subject
2:01:54to
2:01:55but you see that this maximum here it
2:01:59changes as a function of the parameters
2:02:01but it doesn't change in a very uh
2:02:05dramatic fashion as gamma changes as J
2:02:08changes you're going to have to to
2:02:11change your price if you want to make
2:02:13more profits and probably you'll have to
2:02:16uh
2:02:18to do tetan Mo as it's called in
2:02:21economics so you know by trial and error
2:02:24you'll probably converge to the place
2:02:26where you want to be and nothing big
2:02:28happens
2:02:29but now here
2:02:31in this case it's very different because
2:02:34you see that the profit will look like
2:02:38something
2:02:40you do it in red as well so it stops
2:02:44negative if price is below P0 and then
2:02:48it's going to grow linearly
2:02:50because Phi is equal to 1 and then at
2:02:53this point
2:02:54is going to jump down
2:02:57and do something like this
2:03:01and so profit maximization in this case
2:03:04leads you to this angular point
2:03:07but that's very bad because it means
2:03:10that that's what I call it Cliff Edge
2:03:11optimization
2:03:15[Music]
2:03:21because optimization leads you to a
2:03:23point where of instability
2:03:26and if you don't know extremely well the
2:03:28value of J then you're going to miss
2:03:31this point you're going to overshoot for
2:03:33example and as I've said overshooting
2:03:35means that your profit will drop
2:03:37suddenly
2:03:38and also if you want to go back and
2:03:42reverse calls then it's going to be
2:03:44extremely costly because in order to get
2:03:46back your your attendance you need to go
2:03:48too much lower prices
2:03:51so this is a very interesting scenario
2:03:55and you see that there's no precursor
2:03:57that helps you anticipating the problem
2:03:59because Phi here is stuck to one so you
2:04:03you know you think that everything is
2:04:04all right you increase your price and
2:04:06nothing changes
2:04:07until the point where you know something
2:04:10changes and something changes big
2:04:12so there's no way for you to anticipate
2:04:15the point where things are going to go
2:04:17bad but still they are going bad at one
2:04:19point
2:04:20so this this extra scenario compared to
2:04:24the random field I think model is
2:04:26extremely interesting because it shows
2:04:27that in such situations in situations
2:04:30where you can have abrupt changes then
2:04:33the whole idea of assuming that people
2:04:35optimize their profit may lead
2:04:38in this very simple case uh to a cliff
2:04:42Edge but but in other cases maybe the
2:04:45whole economy might because people are
2:04:49um optimizing their profit it might lead
2:04:52the system as a whole close to a point
2:04:55of instability
2:04:56and so this scenario which uh has been
2:04:59promoted by uh several people in the
2:05:02past which sometimes is called
2:05:04self-organized criticality
2:05:14is I think a very exciting idea that
2:05:17comes from people working in statistical
2:05:19physics stuff uh the idea is that
2:05:23complex systems when you try to optimize
2:05:25them
2:05:26very often you you drive them close to
2:05:30an instability and close to a point
2:05:32where they start malfunctioning
2:05:34completely
2:05:35and so this idea that maybe the whole
2:05:37economy because people are striving to
2:05:40optimize their profits
2:05:42is intrinsically unstable intrinsically
2:05:46close to a point where things may go bad
2:05:49so there's there's the paper in the 90s
2:05:53uh
2:05:55you know drawing the attention of
2:05:56economists to this scenario of
2:05:58self-organized criticality and recently
2:06:00this idea has been picked up in
2:06:03particular by students of Mind Jose
2:06:06Moran and myself so this idea that uh
2:06:11that because of complex of the
2:06:13complexity of the system and because of
2:06:15the existence of
2:06:17of many solutions so in this case there
2:06:20are only two solutions and you see that
2:06:22the existence of multiple solutions to
2:06:24the equilibrium equations can lead to a
2:06:28dramatic effects
2:06:32Okay so
2:06:33that's what I wanted to say about
2:06:37this problem and again I think this
2:06:40scenario of
2:06:42of self-organized criticality and
2:06:45distension between optimization and
2:06:47stability optimization and fragility the
2:06:50fact that complex systems are maybe
2:06:52intrinsically fragile when they are
2:06:54close to Optimum is something that you
2:06:57might also see in other situations like
2:07:00sun piles or
2:07:03um
2:07:07other physical systems
2:07:20okay now I want to introduce you to a
2:07:25different
2:07:27set of ideas
2:07:34so I'm going to call here
2:07:37in the filter title Choice Theory
2:07:45so this is this is a very uh large part
2:07:49of the literature in economics or in
2:07:52social sciences is how do people choose
2:07:54before between different options
2:07:57and I've we've already encountered
2:08:00something like this
2:08:01uh in the random field icing model where
2:08:04people choose by comparing their
2:08:07idiosyncratic propensity to do something
2:08:11or there as I said there
2:08:17willingness to pay
2:08:19intrinsic willingness to pay this with
2:08:21hi in the restaurant problem then the
2:08:25choice was simple either they did
2:08:26something or they didn't do didn't do
2:08:29that thing depending on the value of age
2:08:32compared to some threshold okay but we
2:08:35want to generalize that
2:08:37and introduce the a little bit of of
2:08:39possible noise in the in the decisions
2:08:42that people take
2:08:44and so what uh people have introduced is
2:08:48this Choice Theory framework which as
2:08:51you're going to see is very close to
2:08:52things that we know uh from uh
2:08:55statistical mechanics
2:08:57so for the moment I'm going to consider
2:09:00a single agent
2:09:06okay
2:09:09and for a single agent is going to be
2:09:11confronted with a certain number of
2:09:13possible choices
2:09:15Alpha Beta gamma so on so these are
2:09:20possible choices
2:09:22so in the case that I've considered up
2:09:24to now there were only two choices
2:09:29it was a binary decision but
2:09:32let me remove beta because I'm going to
2:09:34use beta for another
2:09:36notation Alpha Gamma and so on
2:09:39but in other cases you can be confronted
2:09:42to multiple choices and in the example
2:09:46I'm going to expand on later the choices
2:09:49will be the neighborhood in a certain
2:09:53city so think of Paris for example with
2:09:56its 20 audio small
2:09:59and maybe you know you have the choice
2:10:02between 20 possible places to live so
2:10:06you can imagine you know whatever
2:10:08situation you want so I'm leaving here
2:10:12completely free the number of options
2:10:14for a certain decision what I'm going to
2:10:17specify is what economists call the
2:10:21utility
2:10:22of each choice and so this is going to
2:10:25be a certain
2:10:27function U of discrete choices alpha or
2:10:31maybe these choices can be even
2:10:32continuous
2:10:34so U of alpha gives you
2:10:37the utility
2:10:40which
2:10:42in my view is not such a greatly defined
2:10:45object but it's the standard object in
2:10:48the
2:10:48in the economics literature the utility
2:10:51of choice
2:10:55Alpha so what is utility well as I said
2:10:58it's not very clear but you can think of
2:11:00it as happiness the satisfaction
2:11:04whatever it's something that allows you
2:11:07to compare your different choices
2:11:09between them and rank them but more than
2:11:13rank them actually give a value to each
2:11:16of your choices and this value is the
2:11:19utility u Alpha okay
2:11:23so now we're going to assume that this
2:11:25agent is faced with a certain number of
2:11:28choices and he can change his mind and
2:11:32you know do something at one point in
2:11:33time and then do something else at
2:11:36another point in time and so what we're
2:11:38going to assume is that
2:11:41the probability
2:11:44W
2:11:45Alpha to gamma
2:11:48is the probability
2:11:53times DT
2:11:57this is the probability
2:11:59that agent
2:12:04switches
2:12:08between
2:12:10alphine gamma
2:12:16between t and t plus DT
2:12:20okay and so we're going to assume that
2:12:22this is done more or less randomly with
2:12:25some rates W Alpha to gamma and the the
2:12:30standard choice in Choice Theory
2:12:33is the following is that W Alpha to
2:12:37gamma and you recognize something that
2:12:39we are used to in physics and there's a
2:12:41good reason
2:12:43in physics to to write that down but
2:12:45it's not so clear why you should do this
2:12:47uh in economics apart from the fact of
2:12:50course that it leads to much simpler
2:12:52calculation but the idea here is to
2:12:56assume that this is given by some rates
2:13:00gamma which is a
2:13:03a rate an object that has a dimension of
2:13:07one over a time so it's an intrinsic
2:13:10rate
2:13:10divided by one
2:13:14plus exponential of beta
2:13:18U Alpha
2:13:20minus U gamma
2:13:26where beta is a certain parameter which
2:13:30is called the intensity of choice
2:13:39so of course in physics beta is the
2:13:41inverse temperature
2:13:42and as I said they are you know uh good
2:13:46reasons based on the uh reversibility of
2:13:50time and um
2:13:52and the canonical type of argument to
2:13:58justify such a choice in the physical
2:14:02world where utility is replaced by
2:14:05energy but in the case of uh sociology
2:14:09as I said it's a it's a reasonable
2:14:10choice but it has it's the choice that's
2:14:13motivated because of the property that
2:14:15I'm going to describe in a second which
2:14:17is called in physics uh detailed balance
2:14:20okay but before going there let's see a
2:14:23little bit what it means it means that
2:14:26if you gamma is less than U Alpha that
2:14:30is if the choice that you considering to
2:14:33make has a lower utility than your
2:14:37present choice
2:14:39then you Alpha minus U gamma is positive
2:14:42exponential of minus of beta times this
2:14:45difference is larger than one and so you
2:14:48tend to reduce
2:14:49uh the probability of going there and in
2:14:53particular
2:14:54if beta is very large
2:14:56then when you gamma is less than U Alpha
2:15:01you never go there
2:15:03so the limit beta goes to Infinity
2:15:07which is the low temperature limit in
2:15:09physics
2:15:10corresponds to rational choices
2:15:19rational in the sense that if the
2:15:21utility of gamma is less than the
2:15:23utility of alpha you don't pick gamma
2:15:26you stick to Alpha or maybe you stick to
2:15:30a better
2:15:31situation so if if the reverse is true
2:15:34if
2:15:36you gamma is greater than U Alpha then
2:15:39this thing is negative and for beta goes
2:15:42to Infinity this is going to go to zero
2:15:44and so every time you consider making a
2:15:47choice
2:15:48then you go for the better solution
2:15:51okay
2:15:53so that's the intuition behind this
2:15:56and so now
2:15:58we want to describe uh
2:16:02the agent as a probabilistic process and
2:16:07so we are going to introduce
2:16:09the probability that the agent has made
2:16:13Choice outside time t
2:16:16this is property
2:16:19find
2:16:21agents
2:16:25in Choice Alpha
2:16:28at T
2:16:31and because of the structure of the of
2:16:34of this model where at you know it's a
2:16:36markovian process whereas each time step
2:16:39the agent may choose to go to another
2:16:41choice in this
2:16:44random fashion
2:16:46then we get that t p Alpha
2:16:50e t
2:16:53is the master equation formalism it's
2:16:55some over gamma of w gamma to Alpha
2:17:00P gamma t
2:17:02minus sum over gamma of w Alpha to gamma
2:17:08P of alpha t
2:17:22so this is a general Master equation the
2:17:25problem with this master equation is
2:17:28that in general
2:17:29for arbitrary choices of w
2:17:32uh is difficult even to describe the
2:17:36stationary state of such an evolution
2:17:40and in a sense
2:17:42you know the whole
2:17:44non-equilibrium physics is plagued by
2:17:47the fact that we don't have the analog
2:17:50of the boltzmann Gibbs measure
2:17:55that allows us to
2:17:57say something general about
2:17:59non-equilibrium
2:18:01systems in physics and it's related to
2:18:04the fact that for arbitrary W's cannot
2:18:08say much about the evolution of these
2:18:10systems
2:18:11but there is one special case where
2:18:13things are easy
2:18:15and this is the case where detailed
2:18:18balance Falls
2:18:22so this is things that you've seen I'm
2:18:24sure many times but let me recall that
2:18:27detail balance because we're going to
2:18:29see something slightly non-trivial about
2:18:31detail balance a little later on
2:18:36so detailed balance means that if
2:18:41so when
2:18:43for all Alpha Gamma there's a relation
2:18:47between
2:18:48the direct process
2:18:55and the inverse process
2:18:59which is that this is equal to
2:19:02exponential of minus beta
2:19:06h of alpha
2:19:08minus h of gamma
2:19:14so
2:19:16there exists
2:19:19the sun function h
2:19:21of alpha
2:19:23such that if this is true for all pairs
2:19:26Alpha Gamma
2:19:28then
2:19:34P of alpha t
2:19:36when T goes to Infinity is given by the
2:19:39boltzmann weight which is 1 over the
2:19:43partition function exponential of minus
2:19:45beta
2:19:46h of alpha
2:19:55so how do you prove this
2:19:58um well the trick to prove this is to
2:20:01introduce
2:20:02uh the analog of the free energy
2:20:05so if you introduce some object f
2:20:09which is given by
2:20:12um data
2:20:15sum over Alpha of p f of t
2:20:20of P of alphan t
2:20:23h of alpha
2:20:26Plus
2:20:27sum over Alpha of P of Alpha and t log
2:20:32of P of Alpha and T
2:20:36so if you introduce this object and
2:20:40um
2:20:42look at the time variation of this
2:20:45object using the master equation you can
2:20:48compute it and you can show I'm not
2:20:50going to do it because it's the standard
2:20:52thing but I'm just reminding you is that
2:20:55this actually
2:20:56always goes down it can only go down
2:20:59with time and so the solution the long
2:21:02time solution is when F reaches the
2:21:06maximum
2:21:07and the maximum of f
2:21:10is given by the boltzmann the minimum
2:21:13all right the minimum of f
2:21:15uh when time goes to Infinity F has to
2:21:18reach a minimum and the minimum is given
2:21:20is given by the Bossman gives measure so
2:21:24this is a side remark I'm sure that many
2:21:27of you have done that calculation but in
2:21:30this case one can show that there's a
2:21:32so-called the apprentov function that
2:21:34goes down with time and thanks to that
2:21:37you can characterize the late time
2:21:40distribution P of alpha for a very
2:21:44general
2:21:45set of W's provided that they obey
2:21:49detailed balance but in the absence of
2:21:51detail balance unfortunately this is not
2:21:53the case and you cannot say much
2:21:56so why is this Choice made by
2:22:00people in Choice Theory useful well
2:22:04because now let's compute W of
2:22:08gamma to Alpha
2:22:11divided by W Alpha to gamma
2:22:14for this particular choice of
2:22:18of transition rate well clearly the
2:22:23the Gammas disappear
2:22:25so we will have
2:22:28one plus exponential of beta
2:22:32U Alpha
2:22:35minus U gamma
2:22:38coming here
2:22:41and
2:22:43um
2:22:43at the numerator
2:22:47it's going to be one
2:22:51plus exponential of beta
2:22:55U gamma
2:22:57minus U
2:22:59Alpha
2:23:03okay
2:23:05um
2:23:07and so the ratio of this
2:23:13so what I can do is to
2:23:16factorize this thing here
2:23:19so it's exponential of beta U gamma
2:23:23minus U Alpha
2:23:26times 1 which comes from here Plus
2:23:30this one here which will get the inverse
2:23:32of this the factor because I factored it
2:23:35out but the inverse of this factor is
2:23:38just this
2:23:39so you see that after factorizing this
2:23:41thing I get the same two terms in the
2:23:44numerator and denominator which means
2:23:47that they cancel out and they only leave
2:23:49this thing
2:23:51so if I
2:23:54compare
2:23:57oh wait I must have done something wrong
2:24:00here
2:24:08two is the same problem with the signs I
2:24:11want to have that H Alpha is minus U
2:24:14H but here it seems that I've got the
2:24:18sign wrong
2:24:19so what did I say wrong here
2:24:22um
2:24:24so I guess I I missed the sign here
2:24:27because
2:24:29um
2:24:30if H is an energy
2:24:33as we used to think in physics
2:24:36then
2:24:38it's when the the energy of the starting
2:24:42site is larger
2:24:44than the energy of the arriving side
2:24:49no no no no all right
2:24:56uh
2:25:06in the ratio
2:25:10I think you just inverted the ratio ah
2:25:13yes sorry yes I'm sorry yes you're right
2:25:16thank you it's because here I've I've
2:25:18wrongly copied uh things here the
2:25:22denominator was gamma to Alpha and not
2:25:24Alpha to gamma thank you so actually it
2:25:26was gamma here
2:25:28and Alpha here
2:25:31and Alpha here and Gamma here thank you
2:25:36so
2:25:38when is gamma
2:25:46okay good so
2:25:49um
2:25:51so you see that in this case
2:25:54by identification if I choose H Alpha
2:25:58equals minus U of alpha
2:26:02then I get the same results and
2:26:05therefore the long time
2:26:09state of the system
2:26:11is that choices are made proportionally
2:26:14to the exponential of the utility key
2:26:17function
2:26:23and so the limit when beta goes to
2:26:24Infinity
2:26:25corresponds to rational choices in the
2:26:29sense that you only
2:26:32take make choices that maximize your
2:26:35utility function all the other choices
2:26:38that have a lower utility will be
2:26:41suppressed in the distribution of your
2:26:43choices
2:26:44so this is a way to extend the idea of
2:26:47rationality to something a little uh
2:26:50milder a little weaker where you tend to
2:26:54make choices that are good for you but
2:26:56you make mistakes or you take irrational
2:26:59choices sometimes you you want to try
2:27:02something else and therefore your your
2:27:05distribution is not entirely focused on
2:27:09the best choice but it's spread out uh
2:27:13over a certain number of different
2:27:15choices
2:27:16so one justification for that for this
2:27:19value of beta which is not Infinity is
2:27:21that often you actually don't exactly
2:27:24know what the utility of a choice is for
2:27:27you and so there's the there's a there's
2:27:30a blurriness in this concept of utility
2:27:32function which is mimicked by
2:27:35introducing the analog of a temperature
2:27:38uh in this case
2:27:43Okay so
2:27:46um
2:27:47up to now nothing
2:27:50very different than what you were
2:27:53used to
2:27:59and
2:28:01now I realized that I've
2:28:05forgot one page my lecture
2:28:10um
2:28:14okay so let me try to
2:28:17reconstruct what I wanted to say
2:28:36foreign
2:28:42so the simplest case is the binary
2:28:44Choice again
2:28:53so if you have a binary choice you can
2:28:55always write U of alpha
2:29:00so Alpha can be either plus one or minus
2:29:05one
2:29:08and in this case
2:29:10um the utility
2:29:12is given by something that you can all
2:29:15always call you zero
2:29:17plus a field h
2:29:21I'm going to call this s
2:29:23equals plus or minus one
2:29:25times s
2:29:34because UI is a function with two uh the
2:29:38the arguments of the function can only
2:29:40take two values and therefore you can
2:29:43always write one of them
2:29:45as u0 plus h and the other one is g0
2:29:49minus page
2:29:52okay
2:29:53so this is the the case where you have a
2:29:56single agent and um
2:29:59no interaction between agents
2:30:02and so this is not super interesting
2:30:04because in particular uh the probability
2:30:07of
2:30:10of s
2:30:11is equal to
2:30:13um
2:30:15exponential of beta
2:30:18HS
2:30:19divided by two hyperbolic costs
2:30:24of beta HF
2:30:27so if H is very large you tend to pick
2:30:30one and if H is very small you tend to
2:30:33take a minus one but what is interesting
2:30:36is that
2:30:37in order to
2:30:39expand the random field I think model
2:30:42we have a way to do this by introducing
2:30:46some temperature uh in the system and so
2:30:50what I'm saying is that if you introduce
2:30:54interactions now between agents as we
2:30:57did before
2:30:58uh the the formalism of choices allows
2:31:01you to think of a slightly more General
2:31:04model where instead of having s i equal
2:31:07fine
2:31:09of
2:31:11um
2:31:12h
2:31:14plus h i
2:31:16plus sum of a g
2:31:18of jig
2:31:20SG
2:31:22you can think of this rule the rule that
2:31:25I've used in the random field Isaac
2:31:27model that I described before as the
2:31:30zero temperature limit
2:31:31of a probability P of the full
2:31:36configuration of f
2:31:38G of S5
2:31:40which is the exponential of minus beta
2:31:45h of all the sis
2:31:50divided by some bed
2:31:53with a h
2:31:56of s i
2:31:58which is equal to the sum over I of
2:32:03capital h plus h i s i
2:32:10minus
2:32:12uh
2:32:13minus sum over I and J of j i j
2:32:19f i s j and with a one-half here
2:32:25so the zero temperature limits
2:32:27of this model
2:32:30picks up
2:32:31the lowest energy states of the model
2:32:35and
2:32:36you can show that if you're at the
2:32:40minimum of this h of s i then it means
2:32:43that all the spins must be in the
2:32:47direction of the field that they are
2:32:49subject to and the field that they're
2:32:52seeing is the sum of the external field
2:32:55idiosyncratic field the random field in
2:32:57the randomizing model plus the field
2:33:00created by
2:33:01the neighbor
2:33:06so if you expand the random field icing
2:33:09model to non-zero temperature that is
2:33:12instead of taking
2:33:14the limits when beta goes to Infinity
2:33:16which corresponds to
2:33:20to this rule here
2:33:22you can redo everything that I've talked
2:33:25about in terms of identifying the
2:33:28different equilibrium states of the
2:33:30system
2:33:31and
2:33:33what you get
2:33:36is now
2:33:40as a function of temperature one over
2:33:43beta
2:33:53and sigma
2:33:55Sigma being if you remember
2:33:58the
2:34:00the width of the density of here so row
2:34:03of H
2:34:05was something like like this
2:34:08okay
2:34:14so in this plane one over Beta Sigma
2:34:18there are actually a whole line of
2:34:21critical points
2:34:24which separates a system with one
2:34:27equilibrium
2:34:33no history this
2:34:40from a phase where there are two
2:34:44equilibrium
2:34:48and hysteresis
2:34:56so if you remember as a
2:35:01in the previous model where
2:35:04the temperature was zero so beta was
2:35:07infinite
2:35:08then we were
2:35:11on that line
2:35:14this is the previous lecture
2:35:21and what we saw in the previous lecture
2:35:22is that indeed
2:35:24there exists the critical ratio of J
2:35:27over Sigma such that if J over Sigma is
2:35:30greater than some value you're in the
2:35:32history resist phase so you see that in
2:35:34this diagram here it corresponds to this
2:35:37uh region whereas if Sigma is strong
2:35:42enough that it's j over Sigma weak
2:35:44enough you recover a continuous
2:35:47evolution
2:35:48so this is the the rule of thumb that we
2:35:51got from the previous model when
2:35:54imitation is weak compared to the
2:35:58heterogeneity of idiosyncratic choices
2:36:02you get a smooth evolution
2:36:05but when the when the educing Radix
2:36:07choices are too narrowly distributed
2:36:10that is if people tend to anyway behave
2:36:14as a single person and have a variety of
2:36:18video synthetic choices that's very
2:36:20limited then as soon as you introduce
2:36:22some heterogeneous some interaction you
2:36:25get these uh is the sudden shift between
2:36:29equilibrium state so we recover that
2:36:32phenomenology here but we see that
2:36:34fortunately it's not restricted to uh
2:36:38zero temperature and the same
2:36:40phenomenology holds even if you consider
2:36:43a more General model where people don't
2:36:45take choices
2:36:47by systematically optimizing their
2:36:50utility function but also allow for some
2:36:53noise so there's a whole region here
2:36:56where the phenomenology that I talked
2:36:58about in the previous lecture
2:37:01holds of course when I when I put J here
2:37:04and here it's not strictly equal to J
2:37:06but it's of older J
2:37:10uh
2:37:12with coefficients that may depend on the
2:37:15on the structure of the lattice and so
2:37:17on
2:37:19but so what I want to tell you about uh
2:37:23before finishing is what happens on the
2:37:25other line which is this line
2:37:29so this line has no heterogeneity
2:37:40so everybody is the same a priori but
2:37:44there is a temperature
2:37:48and what I want to tell you about
2:37:50without going into the mathematics is
2:37:54that if you study this model here
2:38:00the dynamical evolution of the of a
2:38:02population
2:38:03that is interacting through some
2:38:07social pressure but without
2:38:09heterogeneity one can go quite far in
2:38:13the calculation and obtain the following
2:38:16picture so that's that's really what I
2:38:20want to tell you it's not going into the
2:38:22math of the
2:38:24of the model which I won't have time to
2:38:27expand on but just give you the the
2:38:29final results
2:38:39foreign
2:38:48and so what I'm going to tell you is
2:38:49something that I'm sure you've already
2:38:51heard about in other lectures but I want
2:38:54just to give a an extra little twist to
2:38:57to this story so let me uh summarize I'm
2:39:02studying the case where Sigma equals
2:39:04zero
2:39:06beta arbitrary
2:39:12and I'm also studying this model in mean
2:39:15field
2:39:19so jij
2:39:21equals j0 over n
2:39:24with n
2:39:27large
2:39:30but not necessarily infinite
2:39:43so here again what is the making the
2:39:47whole calculation easy is that
2:39:51the whole dynamics of the system instead
2:39:53of having to keep track of all the
2:39:55decisions of every uh individual
2:39:58or the spins of all the uh the direction
2:40:01of all the spins you only need to
2:40:03understand the evolution of mfp
2:40:07which is one of Ren
2:40:14so the old Dynamics is contained in the
2:40:17evolution of this quantity
2:40:34and so if you um
2:40:36look at the master equation that I've
2:40:38erased now in the specific case of this
2:40:41model with no heterogeneity and the mean
2:40:44field interaction but you find that is
2:40:47that mft
2:40:49obeys an effective launch my equation
2:41:10which is the following dmdt
2:41:14equals minus DV VM
2:41:18so
2:41:20the fact that mft obeys an effective
2:41:22multiplying equation means that you can
2:41:23think of M as the position of a
2:41:25fictitious particle which evolves in a
2:41:28sudden fictitious potential which
2:41:31depends on
2:41:32M but also
2:41:35depends on beta
2:41:38and then there's a noise term there's a
2:41:40large one noise term Plus
2:41:42them which
2:41:44is of all the one over square root of n
2:41:53That's The Logical noise
2:41:55and the importance of
2:41:59of the statement here is that for large
2:42:02but finite n
2:42:03the evolution is not strictly
2:42:06deterministic it has a little noise sum
2:42:09but this noise term goes down like one
2:42:11of a square root then
2:42:14so again this is a consequence of the
2:42:18master equation so from the master
2:42:20equation
2:42:28from the master equation describing all
2:42:30the spins
2:42:32you can
2:42:34shrink down this master equation which
2:42:37did as I just said
2:42:39um describes the evolution of the full
2:42:41configuration of all the spins
2:42:43the master equation
2:42:45is the evolution of a probability
2:42:47distribution of over all the
2:42:49configurations you can shrink this
2:42:51description down to a unique object
2:42:54which is mft which is a one-dimensional
2:42:57object and the resulting Evolution which
2:43:00I'm not showing here
2:43:03but it's not super difficult to uh get
2:43:06this this uh equation the the end game
2:43:10is that this is a large line equation
2:43:12with a potential term
2:43:14and
2:43:16in the larger limit it's in the infinite
2:43:19end limits it's a steministic equation
2:43:21but in the large but finite and limit
2:43:23its logical equation
2:43:26so what is V beta of M
2:43:33well it depends on beta and J
2:43:37and what you find is that
2:43:40so here I would need my
2:43:42lecture notes to be absolutely sure but
2:43:46I guess that if beta J
2:43:50is less than two
2:43:52I think it's two but this you have to do
2:43:55check
2:43:57um
2:44:01I'm sorry
2:44:02we cannot see the end of the program I'm
2:44:05sorry
2:44:12thanks
2:44:16so if it's not two it's four this bound
2:44:20but I um it's not very important for
2:44:22what I'm saying
2:44:25um so what I'm describing here is this
2:44:29transition
2:44:30at that point
2:44:32so when beta J is less than two V of M
2:44:36has the
2:44:37unique minimum
2:44:41and it has this shape
2:44:45okay
2:44:47and so if n if capital N was really
2:44:50infinite it would be easy to understand
2:44:52what's going on
2:44:55the particle goes down slope and stops
2:44:58at the minimum of V of M and so what you
2:45:01get is that the magnetization is zero
2:45:04and that's what we know of the ising
2:45:07model at high temperature there's no
2:45:09magnetization the the object is the
2:45:12power magnet and its average
2:45:15magnetization is zero
2:45:17so if n is not uh strictly infinite
2:45:20there are small fluctuations
2:45:22and this comes from the fact that
2:45:25even if spins are independent even if
2:45:29there was no J at all if if you have a
2:45:32free collection of end spins then you
2:45:35know by
2:45:36uh by chance you can have a
2:45:39magnetization
2:45:40an average value of the spins that's
2:45:42slightly non-zero and actually of other
2:45:45square root of M
2:45:50so that's not very interesting but what
2:45:52is more interesting is what happens in
2:45:54the case where beta J
2:45:58is greater than okay two again with
2:46:00maybe four
2:46:03then what you get is
2:46:06the famous
2:46:08Mexican hat potential or uh
2:46:12double well potential so do you see blue
2:46:19hello can you see the blue color yes
2:46:22okay
2:46:23so now what you get is a standard theory
2:46:27of phase transitions where there are two
2:46:29stable
2:46:31States
2:46:33minus M star and plus M star and an
2:46:36unstable State at m equals zero
2:46:41so here
2:46:42this graph is constructed for H
2:46:46equals zero
2:46:50yeah yeah there should be another
2:46:52parameter in the problem which is
2:46:54capital h
2:46:55the external field
2:46:57so I've assumed Sigma to be zero so the
2:47:00little H i's are zero but Capital H
2:47:02might not be zero and so what I'm
2:47:04plotting here is what happens for H
2:47:06equals zero
2:47:12and for H not equal to zero well
2:47:18these two Wells instead of being at the
2:47:20exact same height
2:47:22one is lower
2:47:25than the other
2:47:26so you have something like this
2:47:29so this is you know very close to the
2:47:32phenomenology of the random field Iving
2:47:35model
2:47:35at zero temperature
2:47:37and indeed
2:47:39the phenomenology is the same in this
2:47:42whole uh part of the phase diagram
2:47:45so you can be either at Sigma equals
2:47:47zero as a function of temperature or at
2:47:50zero temperature as a function of Sigma
2:47:51and you find roughly speaking the same
2:47:55uh
2:47:56phenomenology of one stable Point
2:47:59becoming three uh
2:48:01well three solutions uh among which one
2:48:05of them is unstable but the reason I
2:48:07wanted to draw this diagram for you is
2:48:10the following
2:48:12so what I told you about the random
2:48:14field icing model was that if you're on
2:48:17the say on the low branch
2:48:21and you increase H you stay on the low
2:48:23Branch Forever Until the low Branch
2:48:26disappeared
2:48:28so in this graph it means that there's a
2:48:30critical value of H where at one point
2:48:33this Maxi this minimum will be the only
2:48:37one remaining this one just disappears
2:48:39and you flow to the other solution so
2:48:41that corresponds to the jump I talked
2:48:44about
2:48:45so that's what happens
2:48:47at zero temperature and for infinite
2:48:51size systems but as soon as temperature
2:48:53is non-zero
2:48:56we know from the intuition we get about
2:49:00the large one equation we know what's
2:49:02going to happen for example in this case
2:49:04when H equals to zero the system will
2:49:06spend a lot of time in one of the well
2:49:10but because of this random noise
2:49:13even if it's very small
2:49:15there's a small probability that the
2:49:18system is
2:49:19going to be able to cross the barrier
2:49:22and go to and go see the other
2:49:25minimum
2:49:26so in the case of a series of Loops it
2:49:31means that
2:49:33what can happen
2:49:35it's function of H so you remember M had
2:49:38this shape here
2:49:41and then there's a jump
2:49:46it looks like this okay now I told you
2:49:50you're on the low branch and you stay on
2:49:52the low Branch until the low Branch
2:49:55disappears
2:49:59and this is the analog of this minimum
2:50:02here disappearing but actually if the
2:50:05system is not of uh infinite size and if
2:50:09there is some non-zero temperature
2:50:12then there is a small probability that
2:50:14before you reach this point you actually
2:50:17jump
2:50:19and this would correspond to being in
2:50:22this well here
2:50:24and being able to cross an energy
2:50:26barrier
2:50:28thanks to uh this thermal agitation I
2:50:32mean thanks to the equivalent of a
2:50:34thermal agitation
2:50:36so
2:50:37instead of these branches being stable
2:50:40they become metastable
2:50:51and the question is how long will it
2:50:53take
2:50:54for the system to jump even if it's
2:50:58stuck in the in one of the well
2:51:01in the strict mean field case in the
2:51:03strict n going to Infinity case how how
2:51:06long would it jump how would it take for
2:51:07the system to actually realize that it
2:51:10shouldn't be there it should be on the
2:51:12other uh minimum
2:51:16and so from the launcher equation again
2:51:19you can compute this time uh
2:51:22accurately
2:51:25using uh
2:51:29grammar theory of barrier Crossings
2:51:32but the only thing I want you to
2:51:34remember is that
2:51:36if there is a an energy barrier
2:51:42I'm calling B
2:51:45the time to jump
2:51:49switch time
2:51:56is going to be proportional to the
2:51:59exponential of n times B
2:52:03times the coefficient
2:52:08that comes from the coefficient that
2:52:11I've not written here
2:52:13but that's what's really important is
2:52:15that for a mean field system if it's a
2:52:18finite size in principle there is always
2:52:21a possibility to jump from one state to
2:52:24another but the time it needs to do so
2:52:27is exponentially large
2:52:29in n
2:52:30so the N that you see here is actually
2:52:33coming from the end that you see here
2:52:35is the same m
2:52:37and the message is that even if
2:52:40metastability is indeed the reality for
2:52:43finite n in practice as soon as N is a
2:52:47little large say n equals 100
2:52:50then you you never jump
2:52:53okay
2:52:54so that's uh that's a property of the
2:52:56mean field model
2:52:58if you're not in the field and that's
2:53:00really a very big difference I told you
2:53:02that this phenomenology is true even
2:53:05outside the mean field but if you're
2:53:08outside I mean field then these switches
2:53:11uh can take a time which is much less
2:53:13than exponential of n and therefore in
2:53:17real practical conditions uh you should
2:53:20not forget that such events May take
2:53:23place and change a little bit the naive
2:53:26picture I was giving you in terms of
2:53:28these the series of Loops that are
2:53:30followed until uh the point which is
2:53:33called the spinodal point where uh the
2:53:37the equilibrium disappears you can
2:53:40actually jump
2:53:41much before that point depending on the
2:53:45structure of the model
2:53:46so that's what I wanted to tell you
2:53:49today what I want to tell you next time
2:53:52is about
2:53:54um
2:53:55the generalization of
2:53:58the binary Choice Theory to multiple
2:54:02agents
2:54:03so here I've given you a simple example
2:54:05of
2:54:07multiple agents with binary choices we
2:54:10can have a multi-choice multi-agent
2:54:12model
2:54:13so where instead of having Alpha equals
2:54:16plus or minus one alpha can be anything
2:54:19and you can have a lot of interacting uh
2:54:23agents
2:54:25and so what I will show you is that
2:54:29um in some cases you can recover the
2:54:31equivalent of the detail balance rule
2:54:35for the multi-agent case
2:54:38which is not
2:54:40obvious actually this even if you choose
2:54:43this uh
2:54:45um rule that I've given you that I've
2:54:47erased now
2:54:49on the choice theory that you jump from
2:54:52one choice to another
2:54:54for a single agent given by one over one
2:54:57plus exponentials then the fact that the
2:55:00whole system obeyed detail balance is
2:55:03not a given it's something that you need
2:55:05to check
2:55:06and we'll we'll give a Criterion for
2:55:09that
2:55:10and then we'll move to uh the the
2:55:12shelling model of uh uh City segregation
2:55:16aggregate segregation in cities that can
2:55:20be completely solved using the tools of
2:55:23uh statistical mechanics and and then I
2:55:25will end my lecture on that so that's
2:55:28all for today I'm sorry for the last
2:55:30part which I had to improvise a little
2:55:32bit
2:55:34um but I guess it was more or less okay
2:55:37any question
2:55:40yes sorry what is B in your exponential
2:55:44for the time
2:55:46I'm sorry
2:55:49can you read the expression of the time
2:55:52in the expression of the time too
2:55:57this this expression here
2:56:00yes
2:56:02which is
2:56:05oh B
2:56:07e is the barrier this is what I defined
2:56:10maybe you don't see it with a so
2:56:14so this allows me to add one remark is
2:56:18that this time becomes more
2:56:20either because n is small or because B
2:56:23vanishes
2:56:25and actually you see in my little
2:56:27drawing here that as you increase the
2:56:31the magnetic field
2:56:33you not only
2:56:35the balance the the the Minima but you
2:56:38also make the barrier lower
2:56:40so actually at one point and this in at
2:56:44this point here
2:56:46at the speed speed level point the
2:56:48barrier B goes to zero
2:56:51so that's another property of the of the
2:56:54model is that barrier
2:56:58d goes to zero
2:57:00as three solutions
2:57:04pickup one
2:57:14I'm sorry I don't really see where B is
2:57:17I'm color blinded it must be because of
2:57:19this
2:57:20I'm sorry
2:57:22I don't really see where b stands on the
2:57:27on the shin
2:57:31you don't see the B
2:57:35so I should make a larger okay thank you
2:57:37no no it's because I'm kind of blind so
2:57:40it's just because of me thank you
2:57:43okay so let me
2:57:46to maybe a better drawing so B is the
2:57:49height of the barrier you have to cross
2:57:51to go from one well to another
2:57:55so this barrier is actually not
2:57:57symmetric because if the wells are not
2:57:59of the same height the barrier to go
2:58:01from one to the other is not necessarily
2:58:03equal to the barrier to go from uh in
2:58:06the other way around but you see the
2:58:08idea is how much energy you need to
2:58:10borrow to the thermal bath
2:58:12which in this case is the is this one in
2:58:17order to cross the barrier so this is
2:58:19something that a priori should be
2:58:21impossible
2:58:22if there was no uh random term but
2:58:25because of the random term you can
2:58:28exceptionally
2:58:29and the reason it's is it this is the
2:58:33fact that it's exponential in the
2:58:34barrier height means that it's really
2:58:36exceptional uh that you
2:58:39you know gather enough energy from the
2:58:41thermal bath to actually cross the
2:58:43barrier
2:58:45so this is B
2:58:48thank you very much
2:59:00any other question on the lecture on the
2:59:02previous lectures
2:59:03or on the generalized organization of
2:59:07the
2:59:09extra so next week don't forget you have
2:59:11two
2:59:12today sessions
2:59:24yes sorry I was I also have a question
2:59:26yes
2:59:29if we take back the restaurant problem
2:59:32is it possible to generalize generalize
2:59:35a kind of temperature or is it
2:59:37impossible
2:59:39if generalized what sorry sorry the the
2:59:43sound is not very good so I I don't hear
2:59:45very well what you're saying
2:59:46sorry yeah I'm just asking if it's
2:59:48possible to generalize this uh this so
2:59:51so um like temperature to the to the
2:59:54restaurant problem so maybe on the price
2:59:56or something like that so if it's
2:59:58possible to move uh from one phase to
3:00:01the other one for the price without
3:00:03being to the external that you described
3:00:06before
3:00:07yes yes exactly that's that's the point
3:00:09so so
3:00:10in the case of physics this is real
3:00:13temperature in the case of choice Theory
3:00:15this is a little bit of irrationality if
3:00:18you want but the the the mathematics is
3:00:21exactly the same so in the in the
3:00:24restaurant problem you could jump from
3:00:27The High attendance Branch to the low
3:00:29attendance Branch much before the higher
3:00:33sentence Branch disappears because of
3:00:35these so-called activated events this
3:00:38this is a you know going over a barrier
3:00:41is called an activated event
3:00:45um
3:00:46and and you can have exactly the same
3:00:49type of activated events through
3:00:52irrational Behavior if you want
3:00:58okay thanks
3:01:03okay so normally I should be finished
3:01:06with this with writing up this chapter
3:01:08uh soon so I'll send you the PDF uh
3:01:12probably by the end of the weekend
3:01:19foreign
3:01:24well have a good week and uh
3:01:28see you in two weeks
3:01:30thank you very much
3:01:34thank you goodbye
3:01:36bye

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